Parents worry...doesn't matter how old you are. That said you are an adult! You've given them the choice to come with you, and they have said that won't work for them. That's their choice,
I've cruised twice, once with sharing a room with my Aunt (parents in a different room) and the second cruise was just me, myself and I in one room with my parents a few doors down. (dad was jealous cause my balcony was bigger lol) I was able to do what I wanted sometimes with one or both parents tagging along. Given the age difference there were many things I wanted to do they wanted no part of and the same could be said of me. Most nights we simply met up for dinner and compared notes.
I am now investigating solo cruises..the major issue I'm finding is simply being able to afford the airfare to the cruise port and of course the single supplement. However if the stars align and those two things before more doable I'll be on the next boat outta town! it might be a canoe but I'll be on it! For the worrywarts in my family...I will make some accommodations and get the internet package so that I can email them at least each night (probably).
If you feel confidant that you can do a cruise (and the associated headaches that come with travel in general), you've found a cruise line that you like and can afford to do then I say Go for it! I'm sure that your therapist can prescribe something for the anxiety if you need it. I took my meds with me but found that after the plane ride I did not need it (and in fact, would not have needed it then if Dad had not gotten me all worked up, Mom was less than impressed with him).
You might just surprise yourself with how capable you really are!