Well, it seems vague as to what exactly is staying on. Like
64nailhead said, if the exterior light itself is on, there's a short. If you mean the indicator on the dash comes on solid with the parking lights, you have a blown bulb (traditionally). The indicator comes on because, with the parking lights off, it pulls ground through the parking light circuit. When you turn the lights on, that ground goes away and it finds ground through the indicator. Now that you're throwing an LED in the mix, things get complicated because they only work one way and can't backfeed ground. It will require extensive testing beyond throwing parts at it. Get the meter out and go to town.
Why do you want white parking lights in front, anyway? It's 2020, not 1998.