Usually, when I’ve heard people ask “Do you have a problem with authority?” it is coming from someone that is upset that their authority isn’t being followed. I have no problem with legitimate, effective, and compassionate use of authority, but I do have a problem with another type.
For instance, I had been asked more or less this sort of question by a police officer on a few occasions and I did have a problem with them, because they wanted me to do things that they had no legal authority to do. That’s illegitimate authority, and quite frankly, EVERYONE should have a problem with it.
Similarly, some people who have legitimate authority, but don’t know how to weild it effectively. I will ultimately do what that person says if it’s legitimate authority, but I won’t do so without first providing my feedback on why I think it’s a poor choice and what I think a better option would be. As an engineer, I get these kinds of orders often from higher-ups who don’t have the technical knowledge to understand the constraints or problems those decisions create. I believe effective leadership entails more than just making decisions, but a responsibility to be as informed as reasonably possible.
The other thing that really irks me, perhaps even moreso than the others, perhaps because it is so widespread, is when people see authority as a means of control, superiority, a hierarchy or whatever. I may be obligated to still follow the authority who sees it that way, but it definitely bothers me and I have a problem with it. The way I see it, authority is no better and no worse, it is simply a role that someone fills. For instance, at work, as an engineer, my job is to evaluate a problem, come up with solutions, and evaluate them using various criteria and provide those to my boss, it is then their job to weight the evaluations and make a decision. Both roles are valuable, but the responsibility break down is different. I feel similarly about police officers, politicians, parents, whatever. Authority is a role of making decisions and enacting them, and sometimes that means expressing compassion and personal judgment, but it never means that one is superior to another.