How can I transfer ownership of a PS3 account?

You can change pretty much everything associated with the account with notable exceptions being PSN ID and your birth date (used as a secret question in password recovery procedure).

Since authorization is done via email, you need to ask your dad to:

1. Remove his billing info from account in question;
2. Change account e-mail to your e-mail (there will be e-mail change confirmation link to old e-mail, IIRC);
3. Info about which birthdate was used then creating account.

That's it, after this actions you pretty much "own" the account.

P.S. be warned however, that account transfer/selling is prohibited by PSN terms of service:

You may not sell, buy, trade, or otherwise transfer your Online ID, SEN Account or any personal access to SEN through any means or method, including by use of web sites.

If you are unlucky enough it's possible that account will be banned.