Parametrization of the boundary of the Mandelbrot set

Lasse's answer expanded: Let $\psi$ be the map of the exterior of the unit disk onto the exterior of the Mandelbrot set, with Laurent series $$ \psi(w) = w + \sum_{n=0}^\infty b_n w^{-n} = w - \frac{1}{2} + \frac{1}{8} w^{-1} - \frac{1}{4} w^{-2} + \frac{15}{128} w^{-3} + 0 w^{-4} -\frac{47}{1024} w^{-5} + \dots $$ Then of course the boundary of the Mandelbrot set is the image of the unit circle under this map. However, this depends on the (not yet proved) local connectedness of that boundary. Here, for the coefficients $b_n$ there is no known closed form, but they can be computed recursively. Of course we put $w = e^{i\theta}$ and then this is a Fourier series.

I am not quite sure what you are asking. The boundary of the Mandelbrot set certainly is not an analytic curve. In fact, a famous result of Shishikura shows that the boundary of the Mandelbrot set has Hausdorff dimension 2.

Indeed, it is not even known whether the boundary is a curve at all (i.e., locally connected): this is currently probably the most famous conjecture in one-dimensional holomorphic dynamics.

If the Mandelbrot set is locally connected, then there is a natural description of the boundary of the Mandelbrot set (as the boundary values of the Riemann map of the complement of $M$); this is also known to be a natural combinatorial description in many ways. However, as mentioned above, this parametrization is not analytic, or even $C^1$.

To expand on Gerald Edgar's answer, some key phrases for you to look into are "Douady-Hubbard potential" and "external rays."

An external ray is the image of the ray $\arg z = \theta$ for fixed $\theta$ under Gerald's conformal map $\psi$.

The Douady-Hubbard potential is just the harmonic conjugate of the external ray argument: it's the potential for which the external rays are the field lines.

I'm pretty sure it hasn't been proved that $\psi(\zeta)$ is well defined for all $\zeta$ on the unit circle, but I think it is conjectured to be so. (Sometimes this is phrased as saying that the external ray "lands.") However, the external rays at rational angles $2\pi m/n$ are known to land, and moreover, the dynamics at the landing points on the boundary is related to the fraction $m/n$ in a really nice way. (There's an analogy between the doubling map $\theta \mapsto 2\theta$ on the circle and holomorphic maps $z\mapsto z^2+c$, and the dynamics of $\theta$ under the first map are related to the dynamics of the $z\mapsto z^2+c_\theta$ map, where $c_\theta$ is the landing point of the corresponding ray on the boundary of the Mandelbrot set.) Thus, this parametrization of the boundary is indeed an important and natural object (if it is well defined, as conjectured).