my 91 celica's headlights and brake light work but not parking lights

Yes it is possible. Find the wire from the light switch that powers the lights and run new power wire to all lights from that wire. But before I went thru all that trouble check the fuse if blown replace it. If it then blows again remove all light bulbs and replace fuse. If it then does not blow put light bulbs in one at a time till it blows again. Then trouble shoot that socket for a short to ground? If fuse not blown check light sockets for power and ground. You could just have a loose ground wire causing them not to burn. Might take some time to trouble shoot but its probably just a socket or ground issue.