On the parity conjecture in finite-slope families

We generalize to the finite-slope setting several techniques due to Nekovar concerning the parity conjecture for self-dual motives. In particular we show that, for a $p$-adic analytic family, with irreducible base, of symplectic self-dual global Galois representations whose $(\varphi,\Gamma)$-modules at places lying over $p$ satisfy a Panchishkin condition, the validity of the parity conjecture is constant among all specializations that are pure. As an application, we extend some other results of Nekovar for Hilbert modular forms from the ordinary case to the finite-slope case.