The question is: Weather _____ (permit), we can go on a picnic.
In my opinion, I think permitting is correct while the other is wrong because the sentence- weather permitted…- seems to mean that we hold the power over the weather.
But I also found several people use “weather permitted”, I don’t know if that is acceptable or not?
I agree with Alan, I have never heard the term “weather permitted” and would always go with “weather permitting”. Similarly, “if weather is allowed” is also incorrect - who out there is controlling the weather? The weather controls us! So I would say “if the weather allows”.