The T/C Classic Benchmark rifle is basically what the Ruger 10/22T wanted to be. Fantastically accurate right out of the box, and with regular cleaning and maintenance, is as reliable as any other .22. Cleaning and maintenance is a must with the tight match chamber and you'll be better off staying away from cheap .22 ammo like the Winchester Wildcat, etc... CCI Mini Mags, Green Tag, etc...seem to work best and are cleaner.
However, since you said you were on a budget, you have three additional choices that will all make good squirrel, rabbit and plinking rifles. You've already mentioned the Ruger 10/22T. I too love the laminate stocks. Not only do they look great, but they also hold up well under field conditions and you don't have to worry as much about rain and stocks swelling....potentially causing accuracy problems and later splitting or cracking as traditional hardwood stocks are wont to do. The Ruger is plenty accurate for hunting and plinking and there are lots of aftermarket accessories available if you would like to "customize" it to your own likes piece by piece as your "allowance" permits.
Since you like laminate stocks, you might also want to consider the Remington 597. It is available in a very nice laminate stock. It is available in both blue and stainless steel versions. A nice, dependable and accurate rifle, it too would make for a great squirrel gun.
Finally, there is the Savage 64FV. It is one of my favorites and the least expensive of any of the rifles mentioned here. It has a heavy bull barrel, black synthetic stock and comes with weaver bases for mounting a scope or reddot sight. Savage is renowned for the quality and accuracy of their barrels, and the 64FV is no exception! I've shot and handled several dozen of these rifles and currently own two (I also own a T/C Classic Benchmar, Ruger 10/22T and Laminate Remington 597) and would not want to part with either. With the exception of the T/C Classic, the Savages will regularly outshoot both the Ruger and the Remington, using Wal-mart bulk pack Federal 36gr HP ammo, CCI mini-mags, Green Tag, Federal GM, and various Eley loadings. Just about every load I've tried, the Savages always outshoot the Rugers and Remys. Although it doesn't have a laminate stock, the synthetic makes for an ideal field gun, the steel magazines are very simple and reliable, I always carry 4-5 loaded spares with me in the field and I don't have to bother with loose cartridges rattling around. It is a good choice for a first rimfire. The price is right, and it can take a lot of abuse.
If your parents have set a spending cap and you don't have any extra allowance or lawn cutting money to put with it, you might have a better chance convincing your Mom & Dad to buy this rifle for you. Also, before you go shooting, make sure you know the rules to safe gun handling and shooting, be sure of your target and what lays beyond and never point a gun at something you don't want to destroy. If you haven't already done so, why not ask you Dad if he'd like to take a gun safety/hunter safety course with you? Give you guys some "quality" time together, get to hang out with other new hunters/recreational shooters and usually is lots of fun! Hope this helps!