"Can a Taxpayer claim his parent whose only income in TY 2020 is from form SSA 1099 and the amount is $5400?"

There is guidance from the IRS for determining if someone is a qualifying dependent:

You might want to bookmark these links and read the IRS guidance.

Interactive wizards portal for determining dependency:

https://www.irs.gov/help/ita

"What will happen to the stimulus checks that the parent received if he can be claimed as a dependent?"

I think you are confusing Various regulations and provisions in the tax code. A "dependent" for purposes of claiming them on the tax return is not by definition a Qualifying depended for purposes of the stimulus funding rules. Just like support and custody are not the same thing.

"Will the Taxpayer be entitled to receive a stimulus check when he files his 2020 Tax Return?"

Perhaps it would help to review what is really happening:

The funds were paid out as Advanced payment against a projection. The projection used 2018 or 2019 tax returns. But 2020 is the Actuals. You use the 2020 return to reconcile what a person is entitled to, against what they got.

If the person is not a dependent in 2020, then they would be entitled to the payment/credit. That doesn't mean "not being claimed." It means "no longer qualifies as a dependent."

And read the requirements:

https://www.irs.gov/newsroom/economic-impact-payment-information-center-topic-a-eip-eligibility

https://www.irs.gov/coronavirus/second-eip-faqs#Eligibility

One for each EIP.