Lux is lumens per square metre so I think PAR is more like lumens in that sense.

And no, take for example a bulb that is only shooting blue around 430-453 nm and red at 642-662 nm (the sensitivity for cholorphyll a and b). It would have almost no lumens at all but very high PAR, while for example a plain old light bulb would have high lumens and low(er) PAR.

When I calculate PUR-efficiency, PAR and Lux from a bulbs wattage+spectrum+lumens I have a middle step where I get the PAR value for the bulb though..... So a mathematically savvy manufacturer could provide it, and until then I calculate it myself.

But, for a specific bulb, say the compact fluorescent bulbs AH Supply sells, the manufacturer knows, or should know the spectrum for the bulb, so he should easily be able to determine the ratio between PAR and Lux for that bulb. That ratio should be a constant, determined by the spectra. If that were provided, a light meter measuring Lux could be used to determine the PAR reading an any distance from the bulb, in air or in water.

Exactly! If you know the spectral distribution, lux and wattage you can calculate the PAR, PUR etc.