What is the relationship between W, X, Y and P, M retinal ganglion cells?
Short answer
X = P = midget = β
Y = M = parasol = α
W = γ

Background
First off, P and M cells are not equal to parasol and midget cells; on the contrary - P cells are midgets and M cells are considered to be parasols. 

X, Y and W cells are used to denote different retinal ganglion cells in cats, as proposed by experiments Enroth-Gubel &amp; Robson (1966). As you already indicate, these cells have different physiological response properties. X and Y cells have the standard center-surround distribution of their receptive field, X cells having linear additive response properties in center and surround and Y cells nonlinear properties. X cells therefore have predictive responses such that responses can be predicted to be maximal (e.g. light shining on the ON center and no light stimulation on the OFF surround). Y cells tend to respond in more complex ways. W cells do not have a center-surround structure (Bruce et al, 1996).     

Based on morphology, retinal ganglion cells in the cat have been labelled α, β and γ cells, which are thought to correspond to the physiological classes of Y, X and W cells, respectively (Boycott &amp; H. Wässle, 1974).

The P and M division was introduced by De Monasterio &amp; Gouras (1975) based on monkey experiments. P and M refer to the parvocellular and magnocellular connections retinal ganglion cells in monkeys make in the LGN of the brain. Both types of cells have a center-surround receptive field structure like X and Y cells in the  cat. P cells show color opponency and respond to specific wavelengths of light, and are involved in color vision. P cells respond to a range of wavelengths and respond most vigorous when different light intensities illuminate the center and surround (Bruce et al, 1996).    

P cells are considered to be mostly midget ganglion cells, and equivalent to β cells of cat. M cells, on the other hand are considered to be mostly parasol cells and resembling α cells (Kolb, 2001). 

References
- Boycott &amp; H. Wässle, J Physiol (1974); 240(2): 397–419
- Bruce et al, Visual Perception (1996); 3rd ed. Psychology Press
- De Monasterio &amp; Gouras, J Physiol (1975); 251(1): 167–95
- Enroth-Gubel &amp; Robson, J Physiol (1966); 187(3): 517–52
- Kolb H. Webvision (2001). Salt Lake City (UT): University of Utah Health Sciences Center