As a retired public school teacher in NYS after 31 years and adjunct professor at various public and private universities, I agree, although the order of the list in terms of most-to-least frustrating may differ slightly...
1. Administration, including state and federal level as well as district and building level.
2. Bureaucracy, including the teachers' union as well as the entities listed above.
3. Budgets/Waste. This could be tied to the above, but as an orchestral music teacher this drove me nuts. Untold money available for, and in my opinion wasted on, new and useless administration assistant-on-special-assignment types positions, untold thousands on new tech that will be obsolete in a year or two, and if any money left over miraculously filters down to the music department, it ends up going to the marching band so everyone can see shiny new stuff at football games and competitions. Hmmmmm.... thought this was supposed to be an "education" system. My kids played on celli bases that, at times, were literally held together with duct tape when I didn't have time to (poorly) do the needed repairs myself. In the fifteen years I spent in my last district, I think I was allowed to purchase two new bases and two celli.
4. Parents failing to raise their kids or value education. I can't blame the students as much, as mostly they're all products of their upbringings and environments, but yes, it's frustrating when you can't reach the "hard cases" and get them to turn their lives around despite opportunity staring them in the face over and over.
5. Low pay. This is intentionally last on my list, and perhaps my view is skewed by the state in which I lived and taught. While my kids, based upon my salary as a first year teacher, would have been eligible for my first district's need-based "free lunch" program (which always struck me as ironic ... you're paying your teachers so little that they're below the level you think they can provide for their own kids?!?), we didn't use that option. Still, after a couple years in the profession, some moves to larger districts, taking on some extracurricular paid duties, etc., the "long game" of the profession has worked out well. At 31 years, with the required Master's degree and no further applicable post-graduate work, I never hit six figures (although a few of my colleagues managed to do so), but the NYS teachers' retirement system is strong and, between the pension and private retirement accounts we dutifully cultivated over the years, it's worked out well. Other than the first couple years, I never felt grossly underpaid, but I was always playing "the long game" after realizing it would work well for my family. It ain't a job for anyone who wants a lot expensive toys NOW or wants to live a rich-and-famous lifestyle, that's for sure, but it's not indentured servitude, either....at least as I've experienced it.