Simpson makes his definition of parameters and definability in Definition I.2.3 of his book Subsystems of Second Order Arithmetic (can be found here).
On page 5 he says that: "Note that an $L_2$-structure $M$ satisfies the comprehension scheme if and only if $\mathcal{S}_M$ contains all subsets of $|M|$ which are definable over $M$ allowing parameters from $|M|∪\mathcal{S}_M$".
Based on his definition of the comprehension scheme, why do we need to allow parameters from $|M|\cup\mathcal{S}_M$ for the above statement to hold?
Any help is appreciated, thanks.
The comprehension scheme (as usually defined, which agrees with Simpson's definition) allows for formulas with arbitrary numbers of parameters. That is, each axiom in the scheme may have additional free variables. This is why Simpson mentions universal closures in Definition 1.2.4. He discusses this in the first paragraph of page 5.
Therefore, the comprehension scheme corresponds to definability with parameters.