Right now (AU in score may got away w/ FICO-08 if ever implemented) yes having you as an AU will effect your score. How much or good or bad depends on the account.
Does your dad run a high balance? If so it will hurt you. Balances of 9% start to negatively effect your FICO. Has your dad ever been late? Has he been late or other cards (aka likely to be late in the future)?
Basically adding you as an AU will report that card on your CR and it will be included in all calculations.
The age of the account will be calucalated into your oldest account and average account age stats.
Your util % will include the balance and CL from the Home Depot Card.
Any lates on that card will hit your FICO just like they were lates from one of your current accounts.
To keep it simple it would be just as if it was your card.
BTW: If you dad has had the Home Depot card for 7 months and has $0 or low blance now and no lates he likely could get CLI.