The speed of a parametric curve is given by $$\sqrt{(dx/dt)^2 + (dy/dt)^2}$$
Wouldn't this equation just give a very small length of the curve (because of Pythagorean theorem)? Integrating this gives you the length of the curve from t = a to t = b.
In a position graph of 1d motion the tangent line of the curve will give you the velocity. How come in 2d motion the tangent line $dy/dx$ doesn't give you the velocity, but the small length of the curve does?