What are parabolic bundles good for? The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate parabolic bundles (i.e., vector bundles with flags at finitely many points)?  Said differently, how do parabolic bundles arise in nature? 
 A: They arise in analytic theory of differential equations with regular singularities, Riemann Hilbert problem and Painleve equations.
MR1924757
Biswas, Indranil
A criterion for the existence of a flat connection on a parabolic vector bundle. (English summary)
Adv. Geom. 2 (2002), no. 3, 231–241.
MR1488348  Arinkin, D.; Lysenko, S. On the moduli of SL(2)-bundles with connections on P1∖{x1,⋯,x4}. Internat. Math. Res. Notices 1997, no. 19, 983–999.
A: Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set
up of a Riemann surface with cusps. They were trying to generalize the
Narasimhan-Seshadri correspondence on a compact Riemann surface (between
polystable bundles of degree $0$ and unitary representations of the
fundamental group). In the non-compact case, they were able to determine the
missing piece of data - partial flags and weights at each cusp. They
established what is now called the Mehta-Seshadri correspondence. Then they
proceeded to study the moduli space.
Mehta, V. B.; Seshadri, C. S.
Moduli of vector bundles on curves with parabolic structures.
Math. Ann. 248 (1980), no. 3, 205–239.
https://link.springer.com/article/10.1007/BF01420526
Since then, the definition of a parabolic bundle has been clarified (tensor
product with the initial definition is not really computable for instance) and
generalized. This is a long story starting with C.Simpson, I.Biswas, and many
authors. The upshot is that given a scheme $X$, a Cartier divisor $D$, and an
integer $r$, there is a one to one tensor (and Fourier-like) equivalence
between parabolic vector bundles on $(X,D)$ with weights in
$\frac{1}{r}\mathbb Z$ and standard vector bundles on a certain orbifold
$\sqrt[r]{D/X}$, the stack of $r$-th roots of $D$ on $X$. So one can turn your
question in: why are these orbifolds natural ? They were first introduced by
A.Vistoli in relation with Gromov-Witten theory. They also turned out to be
related to the section conjecture (rational points of stack of roots are
Grothendieck's packets in his anabelian letter to Faltings). So parabolic
sheaves - and stack of roots - are ubiquitous. They are also very strongly
related to logarithmic geometry.
A: The paper by Agnihotri and Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, uses a Narasimhan-Seshadri correspondence between parabolic bundles and unitary connections to determine the possible spectrum of a product of two (special) unitary matrices of known spectrum. They start with a triple of unitary matrices with product $1$, N-S relate that to bundles on $\mathbb P^1$ with parabolic structure at three points, classify those bundles as maps of the $\mathbb P^1$ into a Grassmannian, and end up at quantum Schubert calculus of Grassmannians. Maybe not the most obviously natural source of parabolic bundles, but a wonderful application.
