The only time I've ever borrowed one of my parents' credit cards was when I bought gas in high school. Sometimes it was easier to send me on my own, and nobody checks ID when you pay at the pump (at most, you enter a billing zip code). Otherwise, my parents would give me cash or come with me when I wanted to buy things they agreed to pay for.
They did also, when I was in college, make me a cardholder on one of their credit card accounts. I wasn't responsible for the account in any way (so it didn't affect my credit history either positively or negatively), but I had a card with my own name on it. My parents got the statement with their own statements.
That was pretty convenient, largely because I was responsible about it and my parents trusted me. I always asked permission to use it to buy anything we hadn't already agreed upon. But it meant that even when I didn't have a credit card of my own, I was always going to have access to ready money in an emergency. It also just made it easier for my parents to pay for certain things they were willing to pay for.
I actually still have that card, although I only use it every once in a blue moon (occasionally, my parents will tell my DH and I that we can have a meal out on them). I'll probably tell my parents to just take me off the account when the current card expires -- it's not all that useful anymore, and it's one more thing to carry around in my wallet.