Bad capacitor. Remove it and get the specs off it. Purchase a new one on Dad's account but, as I recall Johnstone does not require a mechanical contractor account/license to purchase products. Cash is king with Johnstone. If your locale is different, find a Grainger's store. If you don't have a Grainger's local to you, check with an electrical supply house. Take that cap to them and they will hook you up with a like rated replacement. Does not have to be a Lennox part as long as there is nothing proprietary about it.
Nice job on the troubleshooting.
Q: Does the contactor remain engaged when calling for "cool"? If not, you may need to replace the contactor too.
Check the contactor coil when it is calling for "cool" and the unit is not running. You should see 24VAC at the coil. If this is the case and the contactor is not engaged then you have found the problem; a bad contactor.