Need a bit more information here.
When you try and start the car, does the starter turn over?
Do you hear any clicking noises when you crank the key to start?
Have you tried jumping the car when it doesn't want to start?
How old is the battery? How was it checked?
To test out the park/neutral sensor - try starting in park and neutral - see if anything changes. The sensor itself is on the transaxle, just look for a large, fan shaped module on the side of the transaxle - should be a cable attached to a lever on the outside of the module.
Starter relay is under the hood - should be under one of the covered boxes that line the side of the engine compartment. As they don't sell a 1998 1.6L Corolla here in the US, I'm not sure if the layout is similar to our 8th gen Corollas (1998-2002) or our 7th gen Corollas (1993-1997).
More likely, are intermittent connections at the starter itself (double check the power cables from the battery) and/or the starter contacts.