How to generate sub addresses belonging to one private key
Addresses derived by an HD wallet are intentionally not publicly linkable to one-another (that would be obviously terrible for privacy). Your wallet software will know the addresses are related (ie, it derived them all from the same seed), but a casual observer will not. 

Accordingly, when you pay some bitcoin to a child address, there is no associated activity involving the parent addresses. 


  How to generate sub addresses belonging to one private key in blockchain.com API


You cannot. Each address derived by an HD wallet will have a corresponding, unique private key related to it. The master private key for the wallet will not be able to sign and spend any funds stored at child addresses - you need the child private keys to do that.