How do you have your LED strips wired right now? That is: what turns them on?
The chip you are looking at most likely can't handle the current that your LED strips require. Its' maximum supply rating of 16V is also a little to close to the nominal 13.8 - 14.5 volts present on your vehicle. Look up the term "Load Dump" for a detailed explanation why. And the current rating is definitely too low - but could be boosted with an appropriate buffer (MOSFET or whatever).
However: you may be able to do what you want if your existing signal lights are incandescent bulbs. This trick probably won't work very well with LED signal lights.
Simply connect the LED strip (+) to whatever you normally have it connected to. That might be your park lights or it could be a separate switch that you have installed.
Connect the LED strip (-) lead to the hot lead of the signal light that you want the strip to flash with. You may want to connect a series diode but I don't think that you will need it - these strips usually have LEDs in series in groups of 3 and the allowable reverse voltage on each LED shouldn't be less than 5V.
How this works is like this: the signal light filaments have a quite low resistance while they are not lit. Your LED strips don't consume all that much current - you haven't said how much but I'd be surprised if it were more than 100 mA or so.
When the signal light is NOT lit, the signal lights' hot wire looks like a low-value resistor to ground. Your LED strip lights up at almost full intensity. When the signal light is lit, its' hot lead is at 12V, the same as the LED strip (+) terminal. There is no voltage across the LED strip, so it is off.