Theory: Spatial resolution of FPI with length L and reflectivity R using laser with wavelength Lamda.

1)  Resolution(frequency) =FSR/finesse = [(C/2L)] / [Pi Sqrt[R]/(1-R)] 

where L is the arm length, R is the reflectivity and C is speed light.

2) Resolution (distance) = (Lamda)^2/C* Resolution(frequency)

Explanation: 

FPI uses coherent lasers, and measures frequency or phase difference, and then the results are converted to distance units. They do not directly measure distances. The frequency resolution of the FPI is decided by the separation between the mirrors and its finesse. For example, an FPI using 99% reflective mirrors separated by 1 meter, will have 480 kHz resolution. This would corresponds to 1.8 femtometer resolution in length units.  This is comparable to nuclear dimensions. For eg, the size of a Gold nucleus is ~7.3 fm.  

Application: Working of LIGO interferometer, resolution ~ 10^-19 m 

By designing an FPI with 99.5% mirrors and separated by 4km, the frequency resolution can be further enhanced to ~60 Hz.  For 1064 nm laser, this would corresponds to a distance resolution of (1064 nm)^2/C*60 = 2.2 10^-19 m.  This is precisely the working principle of Laser Interferometer Gravitational Observatory (LIGO). LIGO uses 4 km long FPI with 99.5% reflective mirrors and 1064 nm laser. They have attained displacement sensitivity of 10^-19 m in practice.  Please remember that, 10^-19 m is 10,000 times smaller than the diameter of Gold nucleus.

 Thus, by making indirect measurements in frequency space and converting the results to length units, one attain displacement resolution of not only pm, but 10,000 times better than the diameter of nucleus. 

In FPI we do not measure the length ie we do not measure the picometer displacement rather we measure the frequency displacement (and frequency width) of the incident beam relative to the transmission characteristics of the FPI  ( involving FSR and finesse).  There is no direct measurement of length like with a graduated scale.  It involves two mirrors and larger the separation between the mirrors better will be the frequency resolution for a given finesse and narrower linewidth can be measured. One can then mathematically converts the measured linewidth in terms of length which may be in the pm of fm scale. It is to be noted that a small difference in two wavenumbers corresponds to a length which is much shorter than the lengths related to those wavenumbers.

A simple, arm-waving explanation.  Imagine a Michelson interferometer with one arm on an accurate, stable stage with nm resolution.  Since you know the wavelength of your CW laser (+/-),  you know how many steps the stage must move to sweep out one sinusoid.  Least squares fit the a priori known sinusoid to your measurement.  By subtracting these you get the "noise" in your measurement.  You will be able to 'split' the fringe into thousands of pieces depending on your noise level.  Effectively this gives the dynamic range of your measurement.   A thousandth of lambda is easy.  That gives you sub-nm resolution.  With a lot of work to reduce the ambient noise and stabilize your optics, the real experts lower this by a thousand times more.

I think a key point is that we measure the average position of huge number of atoms on the mirror surface, and also do averaging on the time axis -- picometre-range resolution can be done also with michelson interferometry (have done it), in addition to frequency measurement and fabry-perot (which can go to below pm?)