Solution:
(Ⅰ) Let point $P(x_{1},y_{1})$ on curve $C_{1}$ correspond to point $Q(x,y)$ on curve $C_{2}$,
According to the problem, we have:
$$\begin{cases} x=x_{1} \\ y= \frac {y_{1}}{2} \end{cases},$$
Therefore, the parametric equation of curve $C_2$ is:
$$\begin{cases} x=2\cos \theta \\ y=\sin \theta \end{cases} (\theta \text{ as the parameter}).$$
(Ⅱ) According to the problem, the Cartesian equation of line $l$ is:
$$\sqrt {3}x+2y+m=0,$$
Let the distance from a point $B(2\cos \theta,\sin \theta)$ on curve $C_{2}$ to line $l$ be $d$,
Then
$$d= \frac {|2 \sqrt {3}\cos \theta+2\sin \theta+m|}{ \sqrt {7}}= \frac {|4\sin (\theta+ \frac {\pi}{3})+m|}{ \sqrt {7}},$$
When $m > 0$, $d_{max}= \frac {4+m}{ \sqrt {7}}=2 \sqrt {7}$, solving this gives: $m=10$,
When $m < 0$, $d_{max}= \frac {4-m}{ \sqrt {7}}=2 \sqrt {7}$, solving this gives: $m=-10$,
In conclusion: $m=\pm10$.

Therefore, the final answers are:
(Ⅰ) The parametric equation of curve $C_2$ is $\boxed{x=2\cos \theta, y=\sin \theta}$.
(Ⅱ) The value of $m$ is $\boxed{\pm10}$.