Does $p \equiv q \pmod a \implies p \bmod a = q \bmod a$?

Yes, $\, \bar p:=p\ {\rm mod}\ n =$ least natural $\,\bar p\,$ such that $\, \color{#0a0}{\bar p\equiv p}\pmod{\! n}.\,$ Ditto for $\,\color{#0a0}{\bar q := q}\bmod n.\,$ So $\!\bmod n\!:\ \color{#c00}{p\equiv q}\Rightarrow \color{#0a0}{\bar p\equiv} \color{#c00}{p\equiv q}\,\color{#0a0}{\equiv \bar q}\Rightarrow\bar p\equiv \bar q\,$ by $\,\equiv\,$ is transitive (being an equivalence relation).

More conceptually: $\:p\equiv q\!\iff\!$ they have equal equivalence classes $\,p+n\Bbb Z = q+n\Bbb Z\!\iff $ their equivalence classes have equal least natural element, i.e. equal remainders $\!\bmod n,\,$ thus

$$ \bbox[9px,border:1px solid #c00]{p\equiv q\!\!\!\pmod{\!n}\iff p\bmod n\:\! =\:\! q\bmod n}\qquad$$

It may prove instructive to add further details to the sketched proofs to ensure comprehension (e.g. see here for a direct proof).

Remark $ $ The operational use of mod is often more convenient in computational contexts, whereas the relational use often yields more flexibility in theoretical contexts. The difference amounts to whether it is more convenient to work with canonical normal forms, or arbitrary equivalence classes. For example, imagine how inconvenient it would be to state the laws of fraction arithmetic if one required all fractions to be in normal (reduced) form, i.e. in lowest terms. Instead, it proves very flexible to work with arbitrary equivalent fractions, e.g. to choose both with a common denominator before adding fractions. See here for further discussion of these matters.