PARIS in March 10-17 or June 3-12

There's a great invention you may have heard about - it's called an umbrella! :-) In all seriousness, weather in Paris can be very unpredictable and downright dreary. I was there for two weeks in July and it rained every day - never went over 50F. But I wouldn't let it deter you from a lovely off-peak vacation. Pack for all seasons and your family should be fine. It'll be interesting to see what others think about your potential change of plans...!

If everything else is equal, choose June. It's warmer and there is more daylight. If it rains (it will, but never for long) the rain will be warmer.

There's no guarantee on the weather now or in June. I love visiting Paris and wouldn't want to wait until June - I'd pack to dress in layers and go Saturday. But, only you can decide!

I read that the crowds don't get bad till July. Is that true ? And we will have two mire days in June.

I would choose June, love the longer days of light, nicer to wander around in evenings. Also that gives you more time to plan and do some research.
The crowds are just not a big deal to me,, there are so many ways to avoid long lines( which research will tell you) that its just not a huge issue to me( I usually go June, July, August or September) . I personally am not into walking as much in the rain as some are,, in fact, while there is absolutely no guarantee you won't encounter rain in June, I can guarantee you won't encounter snow.

I would definitely choose June. For the fact that you'll have 2 more days and for the reasons Adam and Pat said. We've been there many times, and early June is not crowded like the rest of summer.