  In this paper we prove long time existence for a large class of fully
nonlinear, reversible and parity preserving Schr\"odinger equations on the one
dimensional torus. We show that for any initial condition even in $x$, regular
enough and of size $\varepsilon$ sufficiently small, the lifespan of the
solution is of order $\varepsilon^{-N}$ for any $N\in\mathbb{N}$ if some non
resonance conditions are fulfilled. After a paralinearization of the equation
we perform several para-differential changes of variables which diagonalize the
system up to a very regularizing term. Once achieved the diagonalization, we
construct modified energies for the solution by means of Birkhoff normal forms
techniques.
