What are the equations of motions for parafoils?
I don't want to sink in rigid body physics and it will be comfortable for me to stay on next level of abstraction.


Then I'm afraid you will stay stuck at your current level.

Your first equation is only a coordinate transform for the airspeed vector.

The reference area is just an arbitrary area. It makes sense to use the same area for all force components, and the wing area has been universally agreed for this. You will note that c$_D$ is quite a bit smaller than c$_L$ - why should the area be different?

When the same area and speed is used, you can calculate with coefficients just as if they were the real forces. This makes comparison between different speeds and different aircraft (or parafoils) much easier.

Regarding the effect of pulling on the risers (yes, that's the proper word): This changes both the load distribution and the local shape of the parafoil and is not easily expressed in a simple equation.