It indicates that she started working more than 15 years ago, hence Depuis, because depuis usually translates to since/ago.

Travailler pendant means works during, travailler pendant les vacances, pendant la grève, ….

I wouldn’t use neither of the propositions as such. What I would say:

Elle a commencé à travailler il y a plus de 15 ans or Elle a commencé à travailler voilà plus de 15 ans

Meaning she may still be working or not, but we don’t focus on that, only on when she STARTED working. The rest of the speech would indicate probably if she’s still working or not.

Using "passé composé" implies that the action is finished, which is quite normal as in this case "starting to work" is a punctual action: right after you started, the action is already finished. Hence I wouldn’t use "depuis" with passé composé, but rather with present, losing the "commencer":

Elle travaille depuis plus de 15 ans

Still implying she started more than 15 years ago, but here without the focus on the starting date but on the duration since then, and that she’s still working.

Using "pendant" can accomodate "passé composé" but not with "commencer" neither I would say:

Elle a travaillé pendant plus de 15 ans

She worked for more than 15 years, but it’s finished now.

That said, if this is a closed choice and you have to fill the gap without touching the rest of the sentence, I would use "depuis" anyway because I consider it "less wrong" than pendant. It will probably be understood the way it’s meant to be understood.

Note that this answer is only valid if the "15 years" was meant to represent a duration or a date. If it was meant to represent the age of the person, that’s a different story, but in this case neither "depuis" nor "pendant" would be a good fit at all…