\begin{aligned} \text{Let } \mathrm{C}_1(h, k) &\text{ be the center of the circle.} \\ \text{The circle touches the x-axis, so its radius is } \mathrm{r}_1 &= \mathrm{k}. \\ \text{Also, the circle touches the circle with center } \mathrm{C}_2(0, 3) &\text{ and radius } r_2 = 2. \\ \text{Therefore, } \left| \mathrm{C}_1 \mathrm{C}_2 \right| &= \mathrm{r}_1 + \mathrm{r}_2 \\ \Rightarrow \sqrt{(h - 0)^2 + (k - 3)^2} &= |k + 2| \\ \Rightarrow h^2 - 10k + 5 &= 0 \\ \text{Change } h \text{ to } x \text{ and } k \text{ to } y, &\text{ then} \\ \Rightarrow x^2 - 10y + 5 &= 0 \\ \text{It is a parabola.} \end{aligned}