Your old Sky+ box will pick up Skysat ( freesat not freeview ) but your parents will need a satellite dish installed.Also I recently cancelled my Sky contract & am receiving Freesat & using the Skycard that's in the Sky+H/D box ( & getting excellent freesat H/D signals)
That's a good attempt at it, well thought out :) it's actually personal video recorder (though I guess strictly I should have said DVR, or digital video recorder)