We have a company website at
example.com which is hosted at a hosting provider - The NS records here are for the providers name server.
Our API is accessible at
api.example.com - this DNS is part of a hosted zone (not a registered domain) on Route53 and it points to a Load Balancer on AWS - The NS records here are AWS's.
I recently inherited this platform and need to add a new record. I can't seem to find the relationship between our registered domain name (at the hosting provider) and our route53 service. I initially thought the NS records at our hosting provider are updated to cater for AWS according to their docs here: https://aws.amazon.com/route53/faqs/
If your domain name is not managed by Route 53, you will need to inform the registrar with whom you registered your domain name to update the name servers for your domain to the ones associated with your hosted zone. If your domain name is managed by Route 53 already, your domain name will be automatically associated with the name servers hosting your zone.
As I mentioned however, the NS records at the hosting provider is their own. Where else can I start looking for the link between our provider and Route 53?