Yes, but the tickets MUST have already been to the parks that day. For example, you can't drop the family off at
Disneyland first thing in the morning, then have one family member get fastpasses in CA. If the tickets haven't been into the parks that day, they won't work.
What a lot of people do is, get into DL. While one person runs and gets FPs in DL, everyone else exits the park, and waits in line for CA. Then, the "runner" catches up with them in line, and they all enter CA together. Then the runner can go get FPs in CA.
I think that method works best for families with older kids, who are really trying to hit as many rides as possible. When we went this summer, our kids were 5 and 2, so that method was just too busy for my family. I'm sure that will be us in a few years, though.