Couple of things come to mind for me. Now I have been whitewater Kayaking with friends who were experienced (one was instructor certified) and the few times we went I was in class II-III rapids. I was new to sport this year. I flipped in the rapids and was able to get out of my little kayak but things happened so fast.....when I came up from underwater I had traveled 200ft. Got banged up good and swallowed alot of water....to the point of almost drowning.
That was from shear panic and crashing waves. Since then I've flipped a few times and stayed calm. But I always had a life vest on and my friends were quick to help out and grab my kayak while I swam for shore...lol.
Now I know what not to do and how to swim in currents.
Having said that:
Need to know a few things. Sounds like your under age (18). Are you fishing alone? What type of kayak do you have? Obvious...do you wear a life vest?
Rule #1NEVER KAYAK ON RIVER ALONE!!!!!
No matter what the current. Things happen.
And its nice to have someone to help you out or call for help if needed.
And just because you have a life vest doesnt mean you cant drown. I know from experience in my first whitewater swim...even with life vest you can still get sucked under by current/hole and if water is moving fast you can still get exhausted from swimming even with a life vest. TRUST ME!!!! I was so tired besides swallowing in water... the swim almost drowned me. Gave me a whole new respect for rivers, going with others, and being safe. Not to mention if you are fishing and have all that tackle and flip and get tangled in the rods or anchor line it wouldn't be pretty lol.
As a parent myself I wont allow my son...who wants to go whitewater kayaking... go until he takes a class on it and learns his roll as well as water safety. And he wouldn't be able to go by himself or with anybody. It would have to be a certified guide/instructor as well as a group.
I don't know the river you speak of but that's a wide river and would most likely require a good swim to shore if you flipped. Current or not. I think your parents are right on this....but......maybe you could talk to them and look into a guide taking you and your dad or mom for that matter. So you can become familiar with the river and find some others that fish it .
Maybe then they would feel comfortable. I'm not sure its what you wanted to hear but.....trust me.....rivers can be dangerous if you aren't familiar with them or don't respect them.