What does it mean for something to be true but not provable in peano arithmetic? Specifically, the Paris-Harrington theorem.
In what sense is it true? True in Peano arithmetic but not provable in Peano arithmetic, or true in some other sense?

The Paris-Harrington theorem is actually the theorem that states that the strengthened [finite] Ramsey theorem is unprovable in first-order Peano Arithmetic.
However, since the Ramsey theorem is provable in second-order arithmetic, and therefore true in the standard model of Peano arithmetic, we can say that the theorem is true in a very deep and concrete sense.
Generally speaking, when we say a statement about the integer is true we mean that it is true in the standard model, which is the model of second-order Peano arithmetic, and it is all the numbers which can be generated by finitely many iteration of the successor function from zero. (Non-standard models of first-order PA exist, and there are non-standard integers which cannot be generated by finite iterations of the successor from zero.)
You may also want to read this thread about the difference between truth and provability.
