We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector µ ∈ [0, 1] 5. The moduli space corresponding to the central weight µ c = (1 2 ,. .. , 1 2) is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the C *-action we obtain a proof of Simpson's foliation conjecture in this case. For each n ≥ 5, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus n points is false.