Use the Implicit function theorem to prove directly that there is a local parameterization for every implicitly defined submanifold.
Hint: Use the decomposition $\mathbb{R}^{n} = ker$ $d_{p}F$ $\times$ $(ker$ $d_{p}F)^{\bot}$
Unfortunately I have no clue how to prove it. In my understanding a submanifold can be locally parameterized in general - in my differential geometry course we have even written down a definition called "A submanifold can locally be parameterized"