Firstly, you won't loose your job over days off for dependent leave (not immediately anyway, over a period of time, potentially you could).
As others have said, you can legally take a reasonable amount of time off to care for child or a dependent (to make arrangements etc). 'Reasonable' time off isn't defined!
An employer can however make you aware of an absence issue when/if it becomes a problem ('too' much time off in their opinion). They also could be meeting with you to discuss if they can help (well better employers would do this).
Secondly, the meeting you had with your boss and HR, was it documented? If so, ask for copies of the notes. It doesn't sound like a disciplinary as no notice was given (yes, your employer can speak to you in a 121 with HR present to discuss anything that concerns them without notice, it is only when things become more formal you would receive notice of any meeting).
So it sounds to me like they're making you aware of an 'issue' with your absence and asking what you plan to do. Like you say, you have limited options so they now know the score. You can only do what you can do, i.e. offer to work back the hours to minimise impact, work from home, share time off with DH etc etc. YES the conv goes on personnel file but they would need to do at least 2 disciplinaries to fairly dismiss you (and you'd need to have more than what they'd deem as reasonable time off...) so far you've had one informal meeting, so long way off from loosing job! Hope I'm making sense?
Remember, it is a legal right you have to this time off- albeit some shit companies and managers will make you feel bad about it.
Which brings us to your Boss and how she is treating you. Three options:
- informally speak to her, coffee or in ask her for 10 min 121 etc and let her know how she is making you feel.
- formal complaint via HR which should be fully investigated (they'd want evidence of bullying behaviour)
- do nothing and look for another job
Personally, I would go with option 1. It's hard to put yourself out there and tell someone how they make you feel but honestly, most of the time the outcome far outweighs the initial dread. It sounds like your boss is inexperienced at dealing with this type of thing to get HR involved so early. Whether she is bullying you is questionable atm. She could be stepping back from your relationship because she's got "too" close in the first place (she is your boss afterall) or because of the absence, and she doesn't know how to handle; who knows. Either way she's handling it appallingly.
Her ignoring you is unacceptable; there is a potential 'bullying' tag here but you need to tell her about her behaviour rather than just say 'you're bullying me'.
I see this type of scenario a lot. Resolution is 99% communication with manager. It annoys me mgrs don't do the right thing.
You sound lovely OP. If you tell them you will give your utmost for your company but that they must give a little back too and cut you some slack, see what happens. If they don't understand, find a new employer who does
Shit that post is long 
HTH