Car won't go into park right away

When I shift into Park it locks, but it doesn't light up the "P" on my dash. Then I can't turn my car off all the way and take the key out because it is not signaling that it is in park. It only happens when it gets could out, below 40 degrees I believe. And I usually just have to sit there and wait until it "signals" or "catches on" and the "P" on my dash lights up. Even if I push the brake and try to shift out of park it won't work.