The most common reason for an old or badly made single phase electric motor to hum but not start, is that the centrifugal switch that operates the starting winding is not making contact. Your motor could be burned out, but it is not the most likely explanation unless it had water enter its windings or was held in a grossly overloaded condition for a time.

You can see a rudimentary circuit diagram for a typical old-style single phase capacitor-start motor here:
http://mcmullanzu.livejournal.com/4538.html

In my experience a motor of the vintage of yours is very likely to be designed as in that diagram, but of course you can easily trace the internal wiring and see. The motor is reversed just by reversing the connections to the starting winding (the one with the capacitor and centrifugal switch in series with it).

You can very easily tell whether your motor has a starting circuit problem. Just disconnect the drive from the motor to the mower (it will be a V belt or a chain: just remove whichever it is), power up the motor so it is humming, and twist the motor's output shaft once fairly briskly. It doesn't matter which direction you twist it in: single phase motors run in either direction with equally poor efficiency. If the motor starts running smoothly and stops humming, it is just a starting circuit problem, not a burned out motor. If it makes you feel any better, I've had single phase motors that I've always started by hand for years, because it was too much trouble to fix the centrifugal switch: no harm was done.

I have never much cared for Crompton Parkinson motors, and would certainly not spend that kind of money to have one repaired. When I need an electric motor I customarily go to the flea market and buy a second-hand one. The usual price for a half horsepower single phase motor seems to be $25-50 depending on luck. The most I've ever paid was $70, but that was a 1.5 hp low speed three phase of advanced design, and I resented paying so much even then.

It might be unlawful for me to tell you how to fix your motor if the starting circuit has stopped working, so I won't, but if you have a resistance meter and about ten minutes to spare you should be able to track it down. Hint: it is usually the centrifugal switch not closing properly when the motor is stopped (it makes a quite audible click just before stopping) but occasionally the starting capacitor has failed, and rarely the starting winding has lost continuity - I've never even had one of those happen to me.