Not true. If your dad is a permanent resident and has been for 5 years he doesn't even need his wife's participation (assuming he's otherwise eligible).
You're referring to the exception that allows permanent residents who've been living with and married to us citizens to apply for us citizenship after being permanent residents for 3 years. Even then the person doesn't need written permission from the other spouse though he would have to show proof that he's been married to and living with her.
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No. The wife signs nothing. What you're referring to is a requirement of 3 years of marriage to apply for citizenship. At this point he can file on his own without need of proving an existing marriage over the past 3 years.
I agree with my colleagues. As long as you have been living in a marital union with the U.S. citizen spouse, who has been a U.S. citizen during all of the 3 years immediately preceding the date of filing the application and up "until examination on the application", you should have no problem.