If I am not mistaken:

PAR is unweighted energy usually between 400nm and 700nm
Lumens / LUX is wieghted against the human eye sensativity to light spectrum (more light in the green wevelenghts = higher lux), and
PUR is weighted against the photosynthetic spectrum (more light in the red and blue wavelengths = higher PUR)

Sure there is some correlation. If I took a standard 32W bulb and overdrove it 4X ODNO and roughly doubled the lumens I could also assume that the PAR roughly doubled as well, although I dont know what it was to start out with.