Is it Free to Watch an Old Episode (On Demand) of Little People Big World

Maybe, it is the content owner that may or may not charge for their On Demand programing.

Most likely only if it's on a channel/network that you subscribe to. 

Whenever you see something On Demand, there will be a dollar sign ($) next to it if it is a PPV. Also when you try to watch it, immediately or after you've downloaded/recorded it, if it is a PPV it will have you confirm purchase before you can enjoy it. This way you know before the fact if there is any charge. And that is up to the content owner on what is free or not through On Demand.

You can also use Parental Controls to set a spending limit for PPVs. Be aware the amount you set is per PPV (not total over time or per bill cycle). Also that limit only affects ordering by remote for that receiver, so you must set for each receiver/HDDVR you have.