"Do this only once each turn" refers only to the ability it is attached to, not other instances of the same ability. In other words, if you control multiple Nykthos Paragons, you will be able to do this once for each of them.

"Do this only once each turn" refers to putting +1/+1 counters on your creatures using the triggered ability. As long as you haven't yet chosen to put +1/+1 counters on your creatures with it, all instances of gaining life will cause Nykthos Paragon's ability to trigger.

Each creature with lifelink dealing combat damage causes a separate life-gaining event. For example, if two creatures you control with lifelink deal combat damage at the same time and you haven't used it yet, Nykthos Paragon's ability will trigger twice. However, if a single creature you control with lifelink deals combat damage to multiple creatures, players, and/or planeswalkers at the same time (perhaps because it has trample or was blocked by more than one creature), the ability will trigger only once.

If a creature you control is dealt lethal damage at the same time that you gain life, it won't receive +1/+1 counters from Nykthos Paragon's ability in time to save it.

If multiple instances of the ability are on the stack, you will be able to put +1/+1 counters for only one of those instances. Once you do that, other instances will do nothing as they resolve.

Once you have chosen to put +1/+1 counters on your creatures, further instances of gaining life will not cause the ability to trigger.