Hi there Apollonia,
Welcome to planDisney and thanks for your question! I always love the chance to answer a question from a fellow Arizonan.
First, I can definitely relate to your situation and have experienced the need to head out of the parks to give members of my family a quick break during the day.
All that is needed for your Mom to re-enter a theme park is her
theme park ticket. She can access the tickets from the
Disneyland mobile app - which it sounds like you're doing for all of your group members' tickets. If you go to the menu section of the app, select "Tickets and Passes," then "Link Tickets and Passes," she can add her individual ticket to her app to show upon re-entry.
Alternatively, as you mentioned, you can most definitely just use a screenshot of your theme park ticket! In fact, a trick I often use during my visits to the Happiest Place on Earth is changing my phone's lock screen background to a screenshot of my ticket so when I enter a park, it's right there and ready to scan.
You'll just need to ensure that whichever ticket was assigned to your Mom at the start of your visit is the ticket she's using for entry into the parks. On first scan of your tickets, a Cast Member will take a quick picture of each member of your party that will be assigned to their ticket and this is what is used on re-entry to ensure the correct person is using their associated ticket.
Thanks again for your question, Apollonia! I hope you enjoy your visit to the Disneyland Resort.
Have a Magical Day,
Greg