Yay a bigger tank!
A heater is more important than a filter in the world of bettas. Bettas don't like much flow anyhow, so you should watch the filter closely anyhow, and see if it is making too much current for your betta, and if it is, you will either need to baffle it or just plain turn it off.
At my local petsmart I get 50wat heaters which are perfect for 10 gallon tanks, and they tend to run about 20 bucks a pop, so if you can weedle your parents into getting you one, or just taking you there so you can buy one, that would be handy.
Or, if you 10 gallon has a hood light (or lights) you might also try putting some heating lights in them. I have one 10 gallon that instead of a water heater has some old bulbs in it from a herp set-up that someone broke down years ago and gave to me, and those help to keep the water warm too. They are not as stable perhaps as the heater would be, but my house is quite warm anyhow, and the betta in the tank certainly seems happy with the arrangement. The lights are also cheaper than a heater, runnning about 2.50 - 3.50 a piece (my tank hood holds two lights).