  We study the valence crossover and strong electron-phonon coupling of
EuPd$_2$Si$_2$ by polarization-resolved Raman spectroscopy. The fully-symmetric
phonon mode shows strongly asymmetric lineshape at low temperature, indicating
Fano-type interaction between this mode and a continuum of electron-hole
excitations. Moreover, the frequency and linewidth of the phonon modes exhibit
anomalies across the valence-crossover temperature, suggesting the coupling
between valence fluctuations and lattice vibration. In particular, two phonon
modes show significantly enhanced Gr\"uneisen parameter, suggesting proximity
to a critical elasticity regime. The relative contribution of the structural
change and valence change to the phonon anomalies is evaluated by density
functional theory calculations.
