  We present a structure theorem for the moduli space R_7 of Prym curves of
genus 7 as a projective bundle over the moduli space of 7-nodal rational
curves. The existence of this parametrization implies the unirationality of R_7
and that of the moduli space of Nikulin surfaces of genus 7, as well as the
rationality of the moduli space of Nikulin surfaces of genus 7 with a
distinguished line. Using the results in genus 7, we then establish that R_8 is
uniruled.
