I give copies of the requirement page from the Academics and Sports book to parents in my den who want to work on specific pins or Belt loops. They check things off and return it to me. This helps them to know that they are really accountable for what they say their child does. You could tell her you need the sheets for record keeping purposes, which is why I ask for them myself.
Also, our cubmaster, when he hands out awards, will ask a scout what they did to earn it (kind of putting them on the spot). Could you have the den leaders ask the boys to bring in projects they did for the pins to "show" the den?
Sometimes you just have to grit your teeth and take the word of the parent. They are really not doing their sons any favors by "pencil-whipping" if in fact that is what they are doing. Oh, and that Tiger will have to save something for when he's a Webelo, or do things over again, when it comes time to earn Activity pins. Make sure that Mom understands that.
I also wondered if maybe this mom homeschooled? Many homeschoolers use scout books etc in their curriculum.
I agree, that's an awful lot to have done since September. Are they brand new scouts, or maybe the Webelo worked on them over the summer?