did you buy a new clicker? not sure if your truck has one, but usually the cars have a intermittent relay that just plugs in. They are $10-$20 maybe. It's the think that goes "click,click,click" when the blinker is on.
Another thing to check is the stereo (if there is an aftermarket stereo) light wire. Your dash lights are hooked into the parking light system and headlight system. i.e. your dash light wire grounds out, your parking light fuse will blow.
My jeep does the same problem, but I'm too lazy to look behind the stereo. All my lights work. Usually if the blinker blinks fast, a bulb is out. Remember, many lighting systems are hot at the light and ground at the switch...a bad ground could easily be the problem. Check out my post, crazy, but it was just a disconnected ground wire.
Read the first and last posts of both.
HOLY CRAP!! Epiphony!!
One of your blinker bulbs is twisted in backwards...(blinker element and parking lamp element in same bulb) if so, your making a connection between your parking lamps and your left blinker. heck, you hit your left blinker and your parking lamps might even blink...or stay solid...they are solid right?
Check the bulbs. if they are corroded, completely drench the socket with contact cleaner from NAPA or an auto parts store and get new bulbs.
I'd bet $5 this is the problem. let me know if this works!