Is he against horses because of cost or potential danger? If you know his objection, you can then prepare an "argument" for your case. If the objection is danger, find out from the farm about the safety of their school horses, their insurance, their rider requirements regarding safety equipment (helmets, etc.) and ask your dad to meet the instructor and tour the farm so he can see for himself and ask questions. Have your mom talk to other parents of kids that take lessons so she can help your argument.
If cost is the objection, can you negotiate earning the money for lessons? When I was young and wanted something considered a luxury, my dad would agree that if I paid half the cost he would match it. I spent 10 months one year when I was 12 making and selling nylon pot scrubbers to finance 2 weeks at a summer camp. I also was babysitting by age 12 and saved that money as well.