Lenders are required to send you an Adverse Action notice any time they deny credit based on data they obtained from your credit report. That notice should give a high level reason for the denial. It usually shows up in the mail shortly after you're denied. Have her keep an eye out for that letter, and if she doesn't get it soon, have her call the bank and ask for it. Or, she could just call the bank and ask them to explain the denial. Sometimes they won't give specific information, but it certainly doesn't hurt to ask.
It's important to note that a credit score doesn't include DTI, that's calculated separately based on income information you provide when you apply. But, as you've discovered, some issuing banks will report a credit card loan for all authorized users, not just the primary on the account. Other banks don't, but it sounds like you've already determined the literal answer to your question, since the loan is showing up on your sister's report.
It's also worth pointing out that "perfect" credit history doesn't inherently mean "great score" - if you have only ever had one or two loans, even if they've had perfect on-time payments, you may still find that your score isn't that high. Especially if utilization on those loans is really high, duration is short, and they're the same type of loan - both credit cards - versus having a diverse history of different loan types.
If your sister no longer wants to use that credit card, she could have your parents call their bank and remove her as an authorized user, which would (likely) cause the account to drop out of any credit score factors based on current accounts.
She may also want to find a local credit union or customer-focused bank and sit down with them to have a discussion about her credit report, if she's having trouble understanding it. Some institutions will offer free counseling services even if you're not actively looking to start new accounts with them. Or, consider enrolling in creditkarma or other online services that provide explanation of your score.