Incoherent assumption of the parton model Consider the scattering process $ep\rightarrow eX$, in the frame of an ultra-relativistic electron, the partons inside the proton are "frozen," and since the time scale of strong interaction is much larger, the partons are non-interacting.
It is also true that the electron will only hit one parton each time, and so this seems identical to a multi-slit interference experiment and one expects interference terms in the scattering cross section.
However, it is assumed that the electron-parton scattering is incoherent. And so the cross section is proportional to the sum of probabilities $\sum_i |{\cal M}_i|^2$ instead of amplitude squared $|\sum_i {\cal M}_i|^2$. Moreover, we also treat parton distribution functions as real-valued probability distributions instead of complex wavefunctions. 
I'm confused at the point that the partons are treated like "classical" point particles, i.e. the incoherent assumption. 
 A: An ultra relativistic electron has a very small wavelength.  The quarks and gluons in the proton have very small energies with respect to this ultra relativistic energy.  Another way of looking at "frozen", is to think of them on a lattice.

It is also true that the electron will only hit one parton each time, and so this seems identical to a multi-slit interference experiment and one expects interference terms in the scattering cross section.

Interference happens when the wavelength of the incident is comensurate with the distances of the targets ( slits in the two slit experiment) . Due to the very small wavelength of the electron,  as far as interference goes, the scattering  is similar to a very high energy gamma ray hitting a crystal: it will either go through or hit one nucleus and induce a nuclear reaction. 
Do not forget we are talking of electromagnetic interactions here. Higher order diagrams that would form quark loops and interferences with the quark and gluon (parton) content of the proton  through the strong interaction are of very small value due to the (1/137)^1/2 of the electromagnetic coupling constant  entering many times in the loops,  depressing the crossection by at least (10^-8). Thus it is a good approximation for the ultra relativistic energies of our experiments. Once though the electrons get an energy that can generate on mass shell elementary particles  the approximation should start failing.
educational:
The reason the parton model strongly advocated by Feynman became history, and the term "parton" a generic name for the quarks and gluons within the nucleons, is that the parton model did not have enough interactions. It predicted momentum transfer distributions that disagreed strongly with  the high energy physics experimental data. The high energy data showed high momentum transfers at impact, reminiscent of the Rutherford scattering of nuclear physics, "hard targets". Eventually also with the three jet  distributions demonstrating the existence of the gluons,  QCD won.   
