An acceptor is a Boolean function which maps a string of symbols to a Boolean value (i.e. one of {true, false}, {yes, no}, or {1, 0}).
A parser is (an implementation of) a function which maps a string of symbols to a (possibly empty) set of structural descriptions.
The empty structural description is assigned if the string is not `grammatical' in terms of a formal grammar (if the formal grammar is empirically correct, then `grammatical' corresponds to `grammatically acceptable to a native speaker').
If the string has just one analysis (parse) in terms of the formal grammar, then one description is assigned.
If the string is structurally ambiguous, then as many structural descriptions are assigned as there are parses.