Is agreeing to a signature that has not been made, in order to prevent a relative from being imprisoned, considered a false testimony, and does it entail sin? And what is the solution in the future if the parents insist on what does not please Allah and His Messenger, and does the Hadith "There are three whom Allah will not look at on the Day of Resurrection" apply to a person in this case?

Perjury is a false testimony used to achieve falsehood. Your affirmation that your signature is authentic when it is not, is considered a lie. If you know that what is stated in the complaint is true, then your affirmation is not considered perjury; otherwise, it is considered perjury. Obedience to parents is only in what is good and permissible, and it is not permissible to obey them in disobedience to Allah.