the problem is at night when you come up behind them, its kinda hard to tell in a mirror and see what the car is and with all the under cover cars now you never know.
That being said there should not be a reason why we can't have both honestly even if it means installing a 2nd set of lights. you can probably get by with a couple of leds mounted into the housing of the light which is not hard to do(drill hole poke led mount through, poke led through). I am not sure exactly what you were saying about turn signals on with the fogs but with either a relay or a 2nd switch it should be able to be done.
I would really consider this more of a show modification, for parking lots only though as you really could cause some confusion while driving. I know it would make me watch my mirror to wonder what is up, and all it takes is someone looking in their mirror a little too long and not watching to road to see something stupid come out like a deer, kid, another car. I have some blue lights in the grills in my hood and infront of the back tires/side exhaust vents that looks cool for night shows but I would never drive with them on. While it looks cool seriously it could cause a big problem for someone else, and then for you. All it would take is one distracted driver to tell the cops that they were trying to figure out why the parking lights are red and I am sure they could easily go back to show you had been pulled over for it, warned, an ignored it. I mean worst case scenario is someone dying, and you being found responsible. Best case is you look cool, do the math there.....