  We examine the pair entanglement in the ground state of finite dimerized
spin-$s$ chains interacting through anisotropic $XY$ couplings immersed in a
transverse magnetic field, by means of a self-consistent pair mean field
approximation. The approach, which makes no a priori assumptions on the pair
states, predicts, for sufficiently low coupling between pairs, $2s$ distinct
dimerized phases for increasing fields below the pair factorizing field,
separated by spin parity breaking phases. The dimerized phases lead to
approximate magnetization and pair entanglement plateaus, while the parity
breaking phases are characterized by weak pair entanglement but non-negligible
entanglement of the pair with the rest of the system. These predictions are
confirmed by the exact results obtained in finite $s=1$ and $s=3/2$ chains. It
is also shown that for increasing values of the spin $s$, the entanglement of
an isolated pair, as measured by the negativity, rapidly saturates in the
anisotropic $XY$ case but increases as $s^{1/2}$ in the $XX$ case, reflecting a
distinct single spin entanglement spectrum.
