Do my parents still have to file if I'm claiming them on my tax returns?

No, if their only income was Social Security they do not have to file UNLESS they had health coverage through the "marketplace" and received a 1095A.  Otherwise they do not need to file.

As for you claiming them, see the criteria under qualifying relatives:

You can claim a child, relative, friend, fiance (etc.) as a dependent on your 2019 taxes as long as they meet the following requirements:

Qualifying child

• They are related to you.

• They cannot be claimed as a dependent by someone else.

• They are a U.S. citizen, resident alien, national, or a Canadian or   Mexican resident.

• They are not filing a joint return with their spouse.

• They are under the age of 19 (or 24 for full-time students).

• No age limit for permanently and totally disabled children

  •      They live with you for more than half the year (exceptions apply).

Qualifying relative

• They don't have to be related to you (despite the name).

• They cannot be claimed as a dependent by someone else.

• They are a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.

• They are not filing a joint return with their spouse.

• They lived with you the entire year.

• They made less than $4200  (not counting Social Security)

• You provided more than half of their financial support. More info

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them.

Related Information:

• Does a dependent have to live with me?

• What does "financially support another person" mean?

• Can I claim a newborn baby?

This is the credit you will get:

CREDIT FOR OTHER DEPENDENTS

https://ttlc.intuit.com/questions/4499708-what-is-the-500-credit-for-other-dependents-family-tax-credit