Parametrizing an algebraic curve

Suppose $C\subset \mathbb{R}^3$ is a path-component of an irreducible real-algebraic curve. Is there a smooth parametrization that covers all but possibly one or two points of $C$? If not, is there a finite collection of parametrizations that cover $C$ collectively? The parametrization doesn't need to be injective as long as it only ever maps finitely many points together.

My best attempt to construct a parametrization would be to construct smooth parametrizations on an open neighborhood of each point where the curve is nonsingular, and then somehow stitch these neighborhood parametrizations together, possible by using Zorn's lemma.

Just a sketch, showing that the answer is affirmative. Skipping many details.

If the curve is non-singular, then it is a smooth $1$-dimensional manifold ( with several connected components). Each component is analytically diffeomorphic to $S^1$ or $\mathbb{R}$. So you do have parametrizations in this case.

If $C$ has some singular points, then it can be desingularized: there exists a smooth curve $\tilde C $ mapping onto $C$. So now reduce to the previous case.