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Can I say,

(a) He drove up / drove on a winding path.

(b) He drove (up) his car on / up to a winding path.
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I'm not sure of which you are suggesting, Vincent.
He drove up a winding path. Okay. Suggests he's going up hill.
He drove on a winding path. Okay.

I agree with Mike, by the way, that if you are driving, a road is better than a path.

He drove up his car on a winding path. -- no.

He drove his car up to a winding path -- this sounds like he drove up some hill, turned off the road, and then took this winding path. It's possible. But it's probably not what you mean.
Comments  
(a) He drove up / drove on a winding path.

Groovy. No problem and probably fun.

(b) He drove (up) his car on / up to a winding path.

He drove up his car? Whaaaaa?
He drove his car up/on a winding path? Sure. But roads are better for driving upon than paths - you know that, Vincent.
 BarbaraPA's reply was promoted to an answer.