The only snow park I know of is Camp Sacramento. It's located on Hwy 50 about 10 or so miles (maybe more, not sure) outside of South Lake Tahoe. I'm also thinking there used to be a sledding/saucer area at or near Heavenly Valley Ski Resort. You might look into that. Also look into Squaw Valley which is a short drive from South Tahoe.
As for budget-friendly things to do - I would suggest Heavenly Village. It's the large, outdoor shopping area located just before the CA/NV stateline (on the CA side). If you're staying at Harrah's you can walk to it. They have an outdoor skating rink, movies, an arcade, shops and restaurants. We love getting coffee at Starbuck's (or ice cream from Cold Stone in the summer) and watching the people go by. There are a few art galleries and some really fun stores for browsing. My favorite store is called "Up Shirt Creek."
If you want to occupy the kids for a while, Harrah's has my kids' favorite arcade downstairs. We post one adult to stay with the kids while the others go up to the casino to make their donation to the Nevada state economy. It's not free but we've found that $20 worth of tokens goes a long way there.
My parent's own a house in Tahoe so we have been going there since 1970! We don't go much anymore in the winter months because a group of ski patrol guys rents the place from December - April. But we always make at least one trip there in the summer. Hope you have fun!
TIA!!
