Petitioning Parents - Inadmissibility?

[bparcon]

I am a recent US citizen and will be filing for my parents who overstayed their visa to adjust their status after overstaying since 2005.
They bought a house in 2013 with their old SSNs and were asked if they were US citizens - they answered yes so as not to lose the house.
Will this deem them inadmissible? Is there a way to go through it?

[Casebycase]

Did she claim to be a US citizen in paper? I mean, did it leave a trace claiming US citizenship on the paperwork through the mortgage company?

Because, if it was verbal, USCIS would need to prove she actually did it in the first place. They can not make an unfavorable decision only on the assumption that she could've said she was a citizen. HOWEVER, if there's a paper or government form such as the ones below like that show that she intentionally claimed to be a US citizen to get government benefits, that's very serious and I would recommend you seek an attorney BEFORE filing the AOS


Which False Claims to U.S. Citizenship Make You Deportable


If you make a false claim to U.S. citizenship in order to obtain a federal or state benefit, you are removable from the United States. The most common false claims to U.S. citizenship occur under the following circumstances:

• registering to vote in a local, state, or federal election when prohibited from doing so
• checking “U.S. citizen” on an I-9 Employment Eligibility Verification form
• claiming to be a U.S. citizen on a student loan application
• attempting to obtain a U.S. passport, and
• stating that you are U.S. citizen in order to obtain any other benefit for which U.S. citizenship is required.