If the Memory Maker is on one of your parents profiles, then all of the downloads will have to be done through their profile. In order to download your photos, you would need to log into the PhotoPass site with their logins.
As long as you have their logins, yes, you could download your photos first. When you download the first photo, it starts a 30 day countdown. When the 30 days are over, you can no longer add photos to the Memory Maker. That won't be a problem in this situation though, since your parents trip will end less than two weeks after your first day in the parks. You can download your photos when ever you are ready to.
Enjoy your trip!