Why can't we make measurements in a photon's rest frame when loop diagrams make measurements possible? It is one of the axioms of special relativity that the photon has no rest frame; light travels at speed c when measured in any inertial frame of reference. As a corollary, it is often said that if one were in a photon's rest frame, infinite time-dilation and length-contraction would make the universe appear (from a photon's perspective) to be unchanging, with zero length in its direction of motion. In other words a measurement would not be possible in the reference frame of a photon, because there would be no time or space in which one could conduct it.
My question is: how does this accord with the fact that parallel-moving photons can interact with each other via loop diagrams[*]? Photons can therefore be used to perform measurements in the photon's rest frame. I suppose an answer may be "the act of measurement changes the photons momentum vector and therefore its co-moving frame is non-inertial" however this can be said of any measurement.
[*] Or can they? I can't find an explicit amplitude to check this (for, say, the photon-photon "box" diagram), however as far as I can tell from the Euler-Heisenberg Lagrangian nonlinear effects should be present for parallel photons.  
 A: 
In other words a measurement would not be possible in the reference frame of a photon, because there would be no time or space in which one could conduct it.

A measurement is not possible in the reference frame (rest frame) of a photon because there is no such frame. That's all there is to it. You can't make a measurement (or do anything else) in a reference frame that doesn't exist. All the stuff you may have heard about "if one were in a photon's rest frame" is just meant to appease non-physicists who have a hard time understanding why there is no such reference frame.
One of the defining conditions of an inertial reference frame is that there exists a proper orthochronous Lorentz transformation which converts that frame into any other inertial reference frame, and vice versa. But for all such transformations, the starting and ending frames are moving at some relative speed less than $c$; however, any photons that may exist will be moving at speed $c$ in both frames. There is no way to use a Lorentz transformation to produce an inertial reference frame that moves at $c$ with respect to any other inertial frame, and relatedly there is no way to use a Lorentz transformation to produce an inertial reference frame in which a photon would be at rest.
Of course, non-collinear photons do interact with each other via loop diagrams (not directly, though). You can calculate the amplitude of this interaction, which is a reference frame-independent quantity, and turn it into a cross section in any inertial frame you like. But you will never find one in which the photon is at rest.
For collinear photons, perhaps you can still calculate an interaction amplitude. I haven't tried so I don't know if it works. But if you try to compute a cross section or any physical quantity from it, you'll find that the only kinematically allowed processes are those in which the end product consists of only collinear photons with the same total energy moving in the same direction. So any such interactions are really just quantum fluctuations, not actual scatterings.
