Honestly, who cares. Take care of family and then look for another job. Anyone who treats you like this, you shouldn’t be staying at the job regardless. This boss is treating the OP like a slave. Having been around colleges, there are no emergencies, no reason someone can’t take 2 weeks off. They can call a temp service and have someone sit at the OP’s desk and answer the phone for 2 weeks if it’s that important, but trust me, University environments are not that important. They can someone else to cover for the OP, but the boss is a lazy jerk and doesn’t want to be bothered with it.
The benefit with this is that likely HR will be annoyed that there’s this exception and will now be bugging the boss to schedule the OP’s vacation time. It could be a good way to get some pressure on the boss without actually going over his head in a way that he can take offense at.
I’d like to see the statistics on the age of the poster versus where they sit on the line ranging from from “There’s a point where you just gotta say FUCK YOU and go about your life” to trying to find a diplomatic way to resolve this.
Contrary to some peoples belief. you don’t have a right to use your vacation whenever you damn well please. That begin said, it sounds like your boss is being a bit of a dick about whole thing; and while it is at the far end of the spectrum (for me anyway), there definitely is a point where you just gotta say FUCK YOU and go about your life. Seriously, at the very least, you have to talk to the approver in person. A lot of people are assholes, but a lot of people are also very reasonable as well. Especially in light of an ailing family member.
But does the company have a right to continually deny you vacation you have accrued until you lose it?
I work for a university as well, and while generally I’m a flexible department head, if any of my employees flat out told me they were taking vacation it wouldn’t go over well. I almost never reject a request, and when I do it’s with very clear explanation. Things like “you don’t have any vacation time, so no” or “There’s this thing that’s happening that I need you to be here for” - and then we work around that. But my department knows I’m flexible and don’t reject requests without reason. I have to ask my boss when I want to take time: generally it’s not a problem, but if it is, I get an explanation.
I will say from what we know it very much sounds like redheadedstepchild’s boss is being unreasonable, and the main reason I think this is in this:
They might request it? Too damn bad. Someone else requested that day first.
ETA: And ailing family sure as hell takes precedence. When my dad was very sick (and then dying) last fall, no one would have considered saying anything, no matter what else needed to be done.
Oh boy. Been there.
After working for the same boss for 19 (23 now) years I asked for time off to go to my FIL’s funeral. We were on a big project, but my work was done. It was nothing WE had to deliver. Not at all. It was basically accepting a venders software. No set dates to do anything.
My FIL died unexpectedly.
After finding out about the death, I did go into work the next morning to redirect email and such and let my boss know what was up and I would be gone for at least a week (I get 10 days for such events). He said that I only could be gone 3 days. I corrected him. He was furious. 
I walked out. Nearly in tears. Fuck it. - No sleep, upset and just lost my wonderful FIL.
When I returned, my boss told me that “You know I’m still your boss, I’m in charge” I told him that I will be talking to his boss about this.
His boss ripped him a new one. Three hour meeting. And my boss apologized again and again.
The proper response to having a death in the family, or needing to see your father that is in a nursing home is “I’m sorry about that, you should see your father. If there is anything we can do to help with this, let us know”
Managers need to manage and anticipate this.
Of course not, and for that matter I think the OP is getting a raw deal. Furthermore, IANAL, but if you attempted to take your vacation but was rejected, then I think you’d have some legal bases for not allowing it to carry over to the next year. On the other hand asking for time off with very short notice and being rejected is understandable too (I’m not saying that’s what the OP did).