What's the best way to paint clear parts?

I've tried hand brushing Tamiya clear red and orange for turn signals
and brake lights and I have to put on so many coats it never looks
realistic.

I've tried airbrushing it and it always seems to run and I've even tried
airbrushing Mr. Color clear but it looks too dark.

What do you master model makers do when it comes to painting these parts?

You might want to try using enamel paint, model master makes a good red, and amber paint. Those are the ones i mainly use, but humbrol and i think tamiya make enamel too. As for tail lights, one or two coats followed by bmf does the trick. With tamiya you might have to apply a couple coats.

i use unthinned tamiya acrylic, works fine for me.

I hand paint or Airburush them using Tamiya Enamels