Absolutely, Deutsche Bahn, SNCF, ÖBB, etc. are fantastic compared to Amtrak.

Prices are very reasonable, too, if you book at least a couple of weeks in advance (and, ideally, make use of promos). My impression is that prices in the UK are a bit higher than in Continental Europe. The BritRail Euro flexi passes are reasonably priced, though.

One-way rentals, OTOH, can be quite expensive. Maybe your parents can limit driving to Austria?

Also, public transportation options in Austria are good. In areas which cannot be reached by train, there is usually a good bus network both at the local level as well as at the regional level (operated by providers such as Postbus). Again, busses usually run at higher frequencies than what you might be used to from the US (whether we're talking Greyhound/Megabus/etc. or the local transportation authority).