Is there a physical reason why not to think that instead of space expanding, all physical constants and parameters are shrinking (including of course the instruments we use to measure the constants) and space is static, or is it a case of Occam's razor?
The expansion of the universe is happening at large-scales. This means that, if you choose two galaxies, they are moving away from each other at a speed proportional to their distance. The space between the Sun and the Earth, for instance, is not expanding. In fact, some galaxies close to us appear blue-shifted due to their peculiar velocities. The physical constants must be very complicated functions of space and time to mimic such a phenomenon. The almost-FRW universe yields the expansion quite naturally. So, in a sense, you are right that it is because of Occam's razor, but, isn't all of "established" physics due to Occam's razor? :)
Whether a quantity such as length is "shrinking" depends on the choice of units of length. If we used a time-dependent unit of length, we could make the numerical value of each length shrink or expand or do anything we like.
But we are using sensible units of length that are "naturally constant". For example, one meter is defined as 1/299,792,458 of a light second (the distance traveled by light in the vacuum in 1 second) and one second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
So we know that the wavelength of some atomic radiation is a constant multiple of one meter. The same is true for other types of atomic radiation because the ratios of wavelength are constant in time. And the same is true for various other lengths such as radii of planets or stars composed of a fixed material at normal pressure: their size is also de facto fixed as a multiple of the wavelength of some atomic radiation.
So as long as we use sensible units of length, and we do, the constancy of the lengths of various things enumerated above is automatically guaranteed. What you describe may be easily achieved by using unnatural time-dependent units of length, however.