The Phone Parts are an inventory item in Problem Sleuth.
Initially, Problem Sleuth cannot use the phone in his office because it is missing some key components; namely the rotary dial and the cord. This is a double whammy because he can neither call for help nor receive calls from hysterical dames who need their problems sleuthed.
As it turns out the phone parts are in the office of PS's chief rival Ace Dick. It isn't until PS helps AD untie himself (the work of treacherous whores) that AD decides to return the phone parts and help PS escape his office.
In the end just getting the phone parts back wasn't enough; reassembling the telephone merely led to some more weird puzzle shit.