  This paper considers the moduli spaces (stacks) of parabolic bundles
(parabolic logarithmic flat bundles with given spectrum, parabolic regular
Higgs bundles) with rank 2 and degree 1 over $\mathbb{P}^1$ with five marked
points. The foliation and stratification structures on these moduli spaces
(stacks) are investigated. In particular, we confirm Simpson's conjecture for
moduli space of parabolic logarithmic flat bundles with certain non-special
weight system.
