Mingez, what I wrote about the illumination lead in the dash is probably not related to your problem if the radio is stock, but the lighting circuit can have wires that are grounded until they get turned on, then they have 12V. This would mean there is a strong possibility that you have a ground on one of the parking light wires or sockets. The blinkers work ok while the lights are off, but when 12v is put on a grounded conductor you would have results like the ones you described.
As recommended above, check the sockets. Also look for corrosion or fraying in all the wiring harnesses and plugs that are around the lights. Best of luck, electrical issues on cars can be a real PITA