You can not. But you can ride a mountain bike!
As others have noted, no motorized vehicles are allowed on Yellow Fork Canyon Trail. This is a heavily used trail for hiking, running, mountain biking, and horseback riding. People also bring dogs (must be on a leash).
Your pedal bike, yes, absolutely. You'll see several riders all ages. Your motor dirt bike, NO.
Yes, I see mountain bikes all over Yellow Fork. I'm not sure if there are dedicated trails for bikes, but it seems like you can pretty much go wherever. I would just recommend being careful as there are also hikers and horseback riders on those trails as well. Having a bell or other noise maker on your bike would probably be helpful.
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