If the parent is already in the United States, what name is on their U.S or foreign document. It might be easier for you to prove that - that is their correct name. But if your parent are still abroad, then put the correct spelling in the k1 application AND include copies of parent birth certificate and your own birth certificate with the wrong spellings. Then write a letter addressing the wrong spellings, if you can have your parent sign an affidavit on the letter wherever they are. Then great.
I say all that to say this. In the future if you need to petition for them, it’s better to kinda clear it up now on the USCIS side, so that it’s already established in their record.
if you keep sweeping it under the carpet using the wrong names, chances are that it may come back to bite you.
But then again, parent/parent name is not the basis of a k1 application so you should be fine. It shouldn’t affect the approval.