Yes you can water your lawn in the afternoon as long as the sun is not too hot.
If the sun is too hot out then you should wait until later on in the evening to water your lawn so that the hot sun does not evaporate the water.
But you could also have your kids play in the sprinkler or let the sprinkler run on you as well to cool off and water the lawn at the same time when it's hot out.
Early morning or late evenings are the best times though to water lawns especially when it's summer and the temperatures are really hot outside.
As long as the temperature outside is 40 F or above it's okay to water the grass.
If the temperature is lower than 40 F then watering the grass would not be a good idea and it won't do much anyway since when it's really cold outside the grass is dormant and not really drinking the water.
So when you water your grass during cold weather the water mostly just goes into the ground and the grass doesn't really benefit.
When the temperatures start to drop to 40 F or below it's time to stop watering the grass and then water the grass again when spring and summer come.
It's also best to water the grass early in the morning and then later in the evening when it's not too hot as the heat can evaporate the water and the grass could actually burn.