Does a level set of a harmonic function have an analytic parametrisation?

Question

Let $U\subset\mathbb{R}^2$ be an open set. Suppose $f(x,y)$ is a harmonic function on $U$. If $\nabla f=(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial x})\neq 0$ for all $(x,y)\in U$, then the zero set of $f$ is a real analytic curve by the Regular Value Theorem. Suppose the curve is connected and closed. I want to know whether I can express the curve as $t\mapsto(x(t),y(t))$, where $x(t)$, $y(t)$ are real analytic.