  We compute the rational cohomology ring of \bar R_2, the (compactified)
moduli space of Prym curves of genus 2. We also recompute the rational
cohomology ring of \bar S_2, the moduli space of spin curves of genus 2,
thereby correcting some errors made in an article by G. Bini and C. Fontanari,
containing a computation of the same ring.
