Turn signal problems can stem from numerous causes and they can be difficult to track down. Many useful Vintage Mustang Forum discussions have been held on this topic; search the VMF website for "turn signal."
Always start with the simplest solution and work up to the hardest.
Electrical wiring in a car is complex, and interrelated, especially in a 30+ year old car that has had many owners of, let's be kind, varying degrees of expertise at repair and maintenance.
First, check your fuses---all of them. Then check the bulbs.
Next, check the bulb and socket connections. Many reproduction parts are not exact matches with the factory originals, and contacts on some aftermarket parking and taillight assemblies may not align correctly with the corresponding contacts on the bulbs. To test, remove the bulb and check for continuity at the socket.
Now you're working up to the more complex. If the trouble is in one of the many wiring harnesses that snake through the car, it could be difficult to pin down. You'll need a wiring diagram (available from many if not most Mustang parts retailers, in the factory assembly manuals or sold as separate booklets) a continuity tester/volt meter, and lots of perserverance. Use the continuity tester to look for a broken wire or a bad ground connection. Use the volt meter to be sure there is power in the wire at junctures.
submitted by 65Ragtop