On the length of time being laid up because of a sprain...I had the same "bemusement" when reading Jack Caldwell's
Mr. Darcy Came To Dinner, in which poor Darcy broke his leg and was stuck at Longbourn for a month. Nowadays no one would be off their feet for longer than a day or two with either injury. I believe it had to do with the state of medical science at that time, and also the state of assistive devices such as ace-bandages (non-existent?) splints/crutches/canes/walkers (the first three existed in cruder form?) and of course, the modes of transportation. Goodness, last year the physical therapists had me up and walking around the day after my hip was replaced! -- putting weight on injured limbs speeds healing and strengthens bones and muscles, but no one believed that in the 19th century, I suppose.
And on the children question -- I suppose attitudes of the 19th gentry differed from ours. I get the feeling from Chapter 2 that Parker is much more devoted to promoting Sanditon as a fashionable seaside spot than he is to his children. In the beginning of the second chapter, the narrator notes that the children were "sweet" -- is that from Parker's POV, or the narrator's? I get the idea that both the Parkers and the Gardiners were able to afford one or more nannies/nurses/governesses, so thought nothing of leaving the children behind and going off on holidays.
At least the Gardiners left the children with relatives when they were going on the Derbyshire jaunt, since that would be a lengthier absence -- I got the idea in P&P that the Christmas visit was only a week or so. I've also read somewhere that Christmas was not nearly as focused on the children as it is in more modern times. Anyway, maybe the Parkers weren't anticipating such a long absence when they were heading to Willingden, either. But there's nothing to indicate they were concerned about the children when they were unexpectedly delayed, that I can see.
Seems that Mrs. Parker isn't the brightest light, either, from the description of her. She just goes where her husband leads.
Charlotte Heywood = Gigi Darcy??