Do you capitalize mom and dad if used as a possession?

No. As in the sentence...

I am going to the store with my mom and my dad.

However, if you were writing the following sentence...

.

I am going to the store with Mom and Dad.

You would capitalize when proper nouns are used in lieu of their actual names.

The above also applies if you're showing possession:

Dad's car had a flat tire. My dad's job is at a bank.