How do I access paragraphs with the JSON API?
2 steps required :) 


create the paragraph(s) by doing a POST request, and receiving the revision ID of the created paragraph,
update the parent entity by assigning the paragraphs' revision ids to the paragraph field (entity revision reference field).




Step 1.

jsonbody = {
    'data': {
        'type': 'paragraph--study_history',
        'attributes': {
            'parent_id': localStorage.getItem('uid'),
            'parent_type': 'user',
            'parent_field_name': 'field_study_history'
        },
        'relationships': {
            'field_school_ref': {
                'data': {
                    'type': 'node--school',
                    'id': this.reg_selectedUni['id']
                }
            }
        }
    }
};

return this.api.apiPost('/jsonapi/paragraph/study_history', {}, jsonbody)


We are using our custom service api (ApiService.ts) which has the apiPost function that does exactly what you'd imagine :) 
So the logic above should be easy to adapt to your own code.

Note that if you want to add more fields (text fields for example, you add them to the attributes section).



Step 2.
Here’s how we update the user, PATCH it to have the created “study history” paragraphs.

const jsonbody = {
    'data': {
        'type': 'user--user',
        'id': uuid,
        'relationships': {
            'field_study_history': {
                'data': [
                    {
                        'type': 'paragraph--study_history',
                        'id': schoolhist_info['data']['id'],
                        'meta': {
                            'target_revision_id': schoolhist_info['data']['attributes']['revision_id'],
                        }
                    },
                    {
                        'type': 'paragraph--study_history',
                        'id': unihist_info['data']['id'],
                        'meta': {
                            'target_revision_id': unihist_info['data']['attributes']['revision_id'],
                        }
                    }
                ]
            }
        }
    }
};
return this.api.apiPatch(this.endpoint.getUsers + '/' + uuid, {}, jsonbody);


Where this.endpoint.getUsers is /jsonapi/user/user.
This should be enough to get it to work, else comment and I can update.