  Let F be a totally real field of degree d and let p be an odd prime which is
totally split in F. We define and study one-dimensional partial eigenvarieties
interpolating Hilbert modular forms over F with weight varying only at a single
place v above p. For these eigenvarieties, we show that methods developed by
Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight
space of sufficiently small radius, the partial eigenvarieties decompose as a
disjoint union of components which are finite over weight space. We apply this
result to prove the parity version of the Bloch--Kato conjecture for finite
slope Hilbert modular forms with trivial central character (under some
assumptions), by reducing to the case of parallel weight 2. As another
consequence of our results on partial eigenvarieties, we show, still under the
assumption that p is totally split in F, that the full (dimension 1 + d)
cuspidal Hilbert modular eigenvariety has the property that many (all, if d is
even) irreducible components contain a classical point with non-critical slopes
and parallel weight 2 (with some character at p whose conductor can be
explicitly bounded), or any other algebraic weight.
