This is my opinion alone! I have no real references to this. 

Aries: Is the cool grandpa/grandma with a motorcycle and lets their grand kids ride with them. 

Taurus: "Ok, I'll let you do that, but I want you to be careful," or "If you have the large DQ shake, you had better eat dinner tonight!"

Gemini: The grandparent who acts like their grand kids think is cool, but they really aren't. Good effort, Gemini! 

Cancer:  Overly cautious. 

Leo: Puts in all the love they can give, but shows their grand kids off to their friends to let them know that Leo has far better kids. .3.  No offense, other signs!!

Virgo: Teaches their kids math and is the one on speed dial when their kids need help with school. 

Libra: Happy with their husband/wife and cherishes every moment with their grand kids by taking weird pictures of them  I can see my mom doing this... 

Scorpio: Protective of their grand kids and will haunt their enemies in their dreams. 

Sagittarius: Ends up alone and adopts some grand kids. 

Capricorn: Would do all the work for their kids, but makes them do their homework under strict rule. 

Aquarius: Burns the homework, lets the kids do what they want to do. 

Pisces: "Lets go get some kittens for the kids!" (Mom/Dad) "No. We don't want them to end up like you as a crazy cat person... Sorry."