I second trying to sit down with your parents and discuss it with them. Choose a good time. Don't ask them when they get back from work or when they seem tired or strained. Maybe help them clean up after Sunday dinner and then ask if you could sit down with them for 10 minutes. Maybe write it all in a letter if you feel you can't get to sit down.
Try to make a compromise with them and show them you are mature and serious about all this. That you are willing to take on extra responsibilities to help.
It took my dad a while to appreciate my riding lessons. My first barn was 35-40 min away but I only went in the summer twice a week because my dad had 5 weeks off. No way my parents were willing to drive down there every weekend during the school year, so I had to content myself with riding in the summer only.
Then I found a place right under my nose practically (15minutes). I started riding all year then but still not quite regularly. I'd stop in December, January and February, only starting again it back in March going every 2 weeks.
Now I still don't go every week because I pay myself (so limited funds), and I'm really busy with school. But around my final years of high school, I realized riding really helped me concentrate and I got better grades. I told my dad that, but he didn't believe me... until the report cards started coming.
It's important to do physical activities for our mind and body, so if the only way you can ride often is if you change barns, I'd seriously suggest you think about it and start browsing the Internet. I know you love your lease, but if you can't ride him often enough, there's little point in showing him because you need practice.
When I was at my first barn, I didn't want to change for a while because I thought there was nothing closer or better. Well I dag so deep I ended up realizing there were tons of barns all around me. I just had to look. And if some don't' have a website, call them anyways. The barn I am at now is amazing, but I put off visiting it for a while because they didn't have a website I could look at. Now I know why. They are so amazing they don't need to advertise. Word of mouth is good enough for them.
Sorry for the book,
I hope some of this helped.