Solution:

For $p=3$, $p+2=5$, $p+4=7$ and these are obviously prime.

For $p>3$, we know that $p$ is not divisible by $3$. The remainder of $p$ when divided by $3$ can be either $1$ or $2$. If it is $1$, then $p+2$ is divisible by $3$; if it is $2$, then $p+4$ is divisible by $3$. Hence $p=3$ is the only solution.