Other mistake: If it was a party line why did Tony Randall get a busy signal when he was calling Doris Day from Rock Hudson's apartment?
Suggested correction: Dialing the number from a party line would get you a busy signal. If you wish to dial the other party you would dial a special code. In the movie it is 793.
Factual error: After taking Jan home, Rex Stetson/Brad Allen goes back to his apartment and calls Jan. Since they share a party line this would be impossible because a party line is a single phone line shared by two people. For Brad to call Jan he would have to call his own phone number and would receive an engaged/busy signal. (00:41:07)
Suggested correction: Brad Allen explains to Jen that they have a code number for their party line. It is 793. She can call him, and he can call her by dialling 793. No need to dial his own number.