What is interesting here is that drinking 'diet soda' actually decreased the odds for girls (though not reaching statistical significance) - that strikes me as an extremely odd finding. Their discussion of 'why' was not very compelling. I have a hard time thinking there's anything causative with such a disparity in sex.
I also noticed that adjusting for some of the larger confounders for autism (maternal age, SES) didn't move the needle much. I would expect there to be a much larger difference between adjusted/not-adjusted OR. Maybe there's other confounders lurking or the adjustment was insufficient.