An apparent example is "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. I do not see a $\Pi^2_2$ equivalent.
One level up (I originally miscounted - thanks Andreas) we seem to have "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. (In particular, note that "The continuum is greater than or equal to some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)
However, I don't actually see a proof that any of the complexity calculations above are optimal. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.