$15k...
That would be about €2.28 by then.
Seriously, let's think. Should be doable but let's have some numbers down.
3 economy tickets to get them to Europe and back at €900 each = €2700
2 rooms for 20 nights at €100 (4* hotels) a piece = €4000
Average of 1.5 sit-down meal per day for 20 days at mid-range restaurants at €25 per person per meal: €2250

Roughly 9000 Euros, meaning $12700ish at today's rates, leaving them with just under €2000 for the rest.
Doable but they need to do either some careful budgeting or make some sacrifices along the way (such as sharing the room with the teenager, which I wouldn't really opt for, unless the teenager is 13-14 and not very mature for their age). Zero backpacking required.
Definitely avoid London and Paris. Things are very expensive there, and they aren't such great places IMHO unless you want to do a stand-alone visit for 3-4 days...Definitely not as part of a large trip.
My suggestions would include Central Europe (Czech Rep/Hun/Svk/Austria), Southern Italy and Spain.