I will be arriving with my family(4) from Paris on the Eurostar on a saturday at 12:38pm with a flight to LAX at 3:15pm, is this enough time?
Probably not. Probably an hour from the train to HRow. Which means you could be at the airport with two hours to go. If flying business or first class, you would just barely make it. No margin for error.
Richard, how about flying open-jaws into Paris and out of London or vice versa? Or if you already have your tickets, would you consider doing just 1 day in London, then 3 in Paris, then the final 2 in London. (From your other question it looks like that's about all the time you have?)
Delays on the ES could well ruin your plans. Try to change either the train or flight time so you will have a larger cushion of time to travel to Heathrow. A recent trip Paris-London arrived 45 min later than scheduled.
It's wise to get to Heathrow a full three hours before a flight across the Atlantic. I'm with Frank and Swan. Take an earlier train.