I have those exact same tail lights and I love em! Unfortunately, I didn't have the same problem, my problem was with them blinking fast when using the turn signal.
I'm not sure if there is a fuse somewhere inside or not, check your manual and it should tell you. Couple questions though, do they work with the brakes and turn signals? The inner two columns(towards the middle of the car) of LED's shouldn't light at this time.
Check the fuse again and see if it's blown, if it is, there is a short in the system. Double check the connections, make sure they are seated properly in the connectors. There should be four connectors total. Two with three wires (one ground, one 12v for parking lamp and one 12v for brake/flash). Two with two wires, one for just lighting inner two columns for parking lights and one for reverse. Make sure the three wire ones are connected to the outermost plugs on the tail light assy. and that you don't have the the reverse and other two wire connecter crossed.
Do you get parking lamps on with the headlight on, or not at all? If this is the case, then it is probably the headlight switch in the dash board. Or possibly a relay although I'd be surprised if there is one for a parking lamp circuit. Again check the manual for that.
Well, I hope this helps, keep us updated and good luck!
-Nick