Question

"Ah, the romantic soul!” said my parents skeptically, worried that my business would fail, that I was behind the times.

Answer 1

Based on these lines, it seems like the parents wish that the speaker had a more reliable profession. The fact that the parents are skeptical indicates that they most likely do not think that book publishing is a profitable pursuit. Moreover, they seemed worried about the fact that book publishing is disappearing, so they believe that the author might be behind the times.

Answer 2

I would think that they were supportive but skeptical. They would want her to succeed but they aren't sure that she will.