  The {\em planar Tur\'{a}n number} of $H$, denoted by $ex_{\mathcal{P}}(n,H)$,
is the maximum number of edges in an $H$-free planar graph. The planar
Tur\'{a}n number of $k\geq 3$ vertex-disjoint union of cycles is a trivial
value $3n-6$. Lan, Shi and Song determine the exact value of
$ex_{\mathcal{P}}(n,2C_3)$. We continue to study planar Tur\'{a}n number of
vertex-disjoint union of cycles and obtain the exact value of
$ex_{\mathcal{P}}(n,H)$, where $H$ is vertex-disjoint union of $C_3$ and $C_4$.
The extremal graphs are also characterized. We also improve the lower bound of
$ex_{\mathcal{P}}(n,2C_k)$ when $k$ is sufficiently large.
