Hey Patches89,
I experienced similar myself at a previous job. The team I was working in got re-shuffled, and I ended up developing anxiety working under a new manager who was unkind, and held unrealistic expectations. The good news is that there is definitely help out there to help manage anxiety if you need it! I'd certainly recommend seeing your GP to start a mental health plan for access to psychological services, or medication if both you and your GP feel it necessary. I can also recommend Headspace's free meditation recordings if you're so inclined, I found it very helpful myself just to take some time to ground myself before re-approaching the day. I also found it helped immensely with my sleeplessness.
With regards to your immediate issue, have you considered approaching your manager while in the company of a friendly colleague? Having that emotional and social support can be incredibly reassuring when it comes to having conversations we'd rather avoid. It might also be worth thinking about whether your line manager is the best person to request personal leave from. Some work places might have roster managers, daily organisers, or business managers who may be more amenable to the conversation, but also more appropriate to direct the question to in the first place.
Best of luck with managing the anxiety, and visiting your parents! Anxiety is definitely something you can overcome.
All the best,
Tim