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Depends on the bank. I was denied a Sears card when I was first starting out because almost all my info matched my dad's. First name didn't match though. Regardless, I was denied for a bankruptcy on my report. I had never filed bankruptcy, but my dad did. Tried to call the bank to explain, but they would not even talk to me because they believed they were right. Not sure how you can get ahead of it, but if you can, do it.

Welcome @Anonymous I'll ping @ShanetheMortgageMan and he'll have answers for you about your questions. Congrats on getting your finances back on track! Good luck!
The title company will perform a public records search and there is a chance this judgment would pop up. If it pops up, your parents would state it doesn't belong to them, and the title company will have them complete a "Statement of Information" that details the past 10 years of their lives (where they've lived, any names they've used, and other information like that). The title company will then use that information to confirm the judgment does not belong to them. Any information that you have which confirms it belongs to you could also expedite the research process by the title company.
You're welcome, hope it doesn't become an issue but if it does at least you know what to expect.