Disclaimer: Nobody on this subreddit can tell you whether or not you have anorexia, as we are not medical professionals. However I can point out and say a few things.
Some red flags from your post
BMI in the "underweight" range.
but not happy with the number on the scale mostly because I'm worried what others would think of it.
However…
Some things that are concerning, that you need help with, that needs addressed (regardless of whether or not you have an eating disorder)
I skip meals, breakfast always and lunch when I'm too lazy to pack one to take to work or school, and then eat oversized meal with lots of snacks when I get home to make up for not eating all day.
Do you skip meals accidentally, or on purpose?
If on accident, many people are the same way. However, that doesn't mean you shouldn't get help for it.
If it's on purpose, then it may be an eating disorder
This past winter I lost a significant amount of weight because when I get busy with school and work, food becomes my last priority but I've yet to put it back on.
It may be that you need to work hard at setting times for when to eat. I have trouble remembering to eat, so if this is what you're experiencing also, then it may be that you need to set times of the day when you eat. For example, breakfast at 7. Snack if needed. Lunch at 12. Snack when needed. Dinner at 6. Snack when needed. Setting an alarm at those times may help you remember to eat.
Something that most people with anorexia don't experience
I don't relish eating, I love it actually
Regardless… (basically a TL;DR)
I don't think this is healthy either way.
You're right. It doesn't sound like it has gotten super serious, but it still is concerning. Forgetting to eat is normal, and you just need to learn to eat, and if setting times to eat helps, do it! As I mentioned, with some of your habits I'm not sure if you're doing it purposefully, or if you're doing it accidentally. It is not healthy to worry about your weight though. It is not healthy to eat nothing all day, then eat a bunch of food for dinner. I doesn't sound like it has become an eating disorder, but it does sound like you need help, support, and treatment to help, so it doesn't become super serious.
Your concerns are valid though, and you do need help. I am glad you are requesting help, and admitting there is a problem. I'd recommend trying to talk to your parents again about your concerns, and see if they will help this time. Even if you don't end up having an eating disorder, that doesn't mean therapists can't help you. I hope your parents listen, and I hope they help you get the help you need. You can also try talking to a school counsellor. You do need help, and it is super important that you have a way to get help, before your eating habits get worse.
Good luck!