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.
http://link.springer.com/article/10.1007%2FBF01420526
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.