The reasons I suggested Capital One and Discover is because they are very easy to get and often easy to get increased credit limits in a limited amount of time, which helps utilization and future apps for better cards.
I started with the CO Journey and will never have to throw it away (I can even convert it to a 1.5% cash back Quicksilver card if I want). It just sits in my drawer until I make one charge a month to keep it active, and helps my average age of accounts tremendously.
The ability for a soft pull credit limit increase more often then AMEX is also another benefit. It didn't take often more than a chat with someone at CO's executive office or a Facebook message.
No doubt, AMEX is a better issuer than CO, but AMEX likes to see history and within a year of having the Journey and a good credit report/history, there's no doubt (well, YMMV
) one could get approved for a BCE or BCP as I did.
I don't think there's any reason the OP shouldn't get their own card, such as a Discover IT or CO Journey, AND be placed as an AU on their parent's AMEX card. This will further his chances of getting an AMEX card when he applies for his own revolver and help the average age of accounts.
Bottom line, this card is not forever. It is to get your credit history started and hopefully the ability to keep forever to enhance your age of accounts if you grow more into the "game."