The result (Ed: as first stated) is not true in general. But first of all, we give a result along the lines of the question which is true. (Ed: Conveniently, it answers the revised question.)
Let $L$ be a language which has a finite or denumerably infinite number of non-logical symbols, and let $T$ be a theory over $L$.
Let $M$ be a model of $T$ of cardinality $\kappa$, where $\kappa$ is infinite.
Then $M$ has $2^\kappa>\kappa$ subsets.
But there are only $\kappa$ expressions of the form $F(a_1,a_2,\dots,a_n,y)$, where $F(x_1,x_2,\dots,x_n,y)$ is a formula of the language, and the $a_i$ are (names of) elements of $M$.
Thus there are no more than $\kappa$ subsets of $M$ that are definable using parameters. There are in fact exactly $\kappa$. We use the formula $y=x$. If $a$ is (the name of) an element of $M$, then $\{a\}$ is definable by parameters.
The analysis can be extended to uncountable languages, by putting a suitable lower bound on the cardinality of the model $M$.
Counterexample (Ed: for the original question): Look at the language $L$ that has a predicate symbol for every subset of $\mathbb{N}$, and let $T$ be the theory over this language whose axioms are all sentences true in $\mathbb{N}$, with the natural interpretation for each predicate symbol. Then every subset of $\mathbb{N}$ is definable, even without parameters. For if $P_A$ is the predicate symbol corresponding to $A \subseteq \mathbb{N}$, then the formula $P_A(y)$ defines $A$.
Admittedly, $L$ is a rather large language. But such languages can be useful in Model Theory, for example in constructing non-standard models of analysis in which every "normal" function and relation on $\mathbb{R}$ can be automatically extended to the non-standard model.