get good grades or help out a lot at the house and if he still says no then go for a cheaper alternative (liek platnum or gold)
How to convince my dad to get me chess.com diamond?

Get to 2500, and then argue it is necessary for your professional growth. Till 2500 use the free ressources that are available everywhere on the net.
Get to 2500? like if it that was easy as you said. 2500 don't need chesscom membership. They may even find coach notations as silly, unlike low rated players like me, they're very useful and tell us to know where we blundered.
Guys my dad said he'd subscribe if it doesn't interfere with my school work!