Since we know that the distance that I traveled to the park and the distance that I traveled to get back home are the same, and that distance $ = $ rate $ \times $ time, we have that \begin{align*} & (x^2)(3) = (16 - 4x)(4) \\ \Rightarrow\qquad & 3x^2 = 64 - 16x \\ \Rightarrow\qquad & 3x^2 + 16x - 64 = 0 \\ \Rightarrow\qquad & (3x - 8)(x + 8) = 0.\end{align*}Solving this equation, we get the solutions $ x = \frac{8}{3} $ and $ x = -8 $. Since $ x $ must be positive, we have $ x = \boxed{\frac{8}{3}} $.