詳しくはリファレンス見てね

https://ja.wp-api.org/reference/categories/



curlコマンドで取得してみる
例)

$ curl -X OPTIONS -i http://demo.wp-api.org/wp-json/wp/v2/categories

結果)
[{“id”:11,”count”:9,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/banner\/”,”name”:”banner”,”slug”:”banner”,”taxonomy”:”category”,”parent”:0,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/11″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=11″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:43,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/fishing-type\/”,”name”:”Fishing Type”,”slug”:”fishing-type”,”taxonomy”:”category”,”parent”:0,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/43″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=43″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:12,”count”:33,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/news\/”,”name”:”news”,”slug”:”news”,”taxonomy”:”category”,”parent”:0,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/12″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=12″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:44,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/fishing-type\/type1\/”,”name”:”Type1″,”slug”:”type1″,”taxonomy”:”category”,”parent”:43,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/44″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/43″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=44″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:45,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/fishing-type\/type2\/”,”name”:”Type2″,”slug”:”type2″,”taxonomy”:”category”,”parent”:43,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/45″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/43″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=45″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:46,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/fishing-type\/type3\/”,”name”:”Type3″,”slug”:”type3″,”taxonomy”:”category”,”parent”:43,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/46″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/43″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=46″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:1,”count”:28,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/uncategorized\/”,”name”:”Uncategorized”,”slug”:”uncategorized”,”taxonomy”:”category”,”parent”:0,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/1″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=1″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:13,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/%e3%82%a8%e3%83%aa%e3%82%a2\/”,”name”:”\u30a8\u30ea\u30a2″,”slug”:”%e3%82%a8%e3%83%aa%e3%82%a2″,”taxonomy”:”category”,”parent”:0,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/13″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=13″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:36,”count”:3,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/top_page\/slide1\/”,”name”:”\u30b9\u30e9\u30a4\u30c91″,”slug”:”slide1″,”taxonomy”:”category”,”parent”:35,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/36″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/35″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=36″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:37,”count”:3,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/top_page\/slide2\/”,”name”:”\u30b9\u30e9\u30a4\u30c92″,”slug”:”slide2″,”taxonomy”:”category”,”parent”:35,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/37″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/35″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=37″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}}]

・親カテゴリをもつカテゴリ(子)を取得
例えば画像のFishing Type(id:43)に紐づく小カテゴリ(Type1、Type2、Type3)を取得する

例)

$ curl -X OPTIONS -i http://demo.wp-api.org/wp-json/wp/v2/categories?parent=43

結果)
[{“id”:44,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/fishing-type\/type1\/”,”name”:”Type1″,”slug”:”type1″,”taxonomy”:”category”,”parent”:43,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/44″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/43″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=44″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:45,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/fishing-type\/type2\/”,”name”:”Type2″,”slug”:”type2″,”taxonomy”:”category”,”parent”:43,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/45″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/43″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=45″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}},{“id”:46,”count”:0,”description”:””,”link”:”http:\/\/demo.wp-api.org\/category\/fishing-type\/type3\/”,”name”:”Type3″,”slug”:”type3″,”taxonomy”:”category”,”parent”:43,”meta”:[],”_links”:{“self”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/46″}],”collection”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories”}],”about”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/taxonomies\/category”}],”up”:[{“embeddable”:true,”href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/categories\/43″}],”wp:post_type”:[{“href”:”http:\/\/demo.wp-api.org\/wp-json\/wp\/v2\/posts?categories=46″}],”curies”:[{“name”:”wp”,”href”:”https:\/\/api.w.org\/{rel}”,”templated”:true}]}}]

実用したのは上記だけなのでとりあえず備忘録的に記載