Lux is lumens per square metre so I think PAR is more like lumens in that sense.Just a thought: doesn't every bulb have a fixed relationship between lux and PAR, based on the spectrum produced by the bulb? That would be a number, like .26 PAR to Lux, and of course it would apply only to new bulbs. Is that right? If so, we should urge bulb manufacturers to provide that for every bulb.
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.