Does the language of Regular Expressions need a push down automata to parse it?
You are correct. It is easy to show that the syntax of regular expressions is not regular using standard techniques.

One possibility is to use a homomorphism (which $\mathrm{REG}$ is closed against) to get rid of all symbols but the parentheses, which leaves you with the Dyck language which is well-known to be non-regular. If in doubt, use the Pumping lemma on $(^p)^p$. 

That said, you probably do not want to code a PDA by hand. Consider using a parser generator like ANTLR or byacc. If, on the other hand, you want to investigate the parsing of languages by programming parsers yourself, you should continue with other basic parsing algorithms such as CYK, Earley, recursive descent and LR.