Find the equation of the parabola that satisfies the following conditions: Vertex $(0, 0)$,passing through $(5, 2)$,and symmetric with respect to the $y$-axis.
- A$x^{2} = \frac{25}{2} y$
- B$2x^{2} = 25y$
- C$x^{2} = 25y$
- D$y^{2} = \frac{4}{25} x$
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