It's possible that the family had something valuable physically taken away from them, but I don't think the expression necessarily implies this. "Quitar la paletita" is not that common, but here are a couple of examples of how it can be used:
"no me ire como un niño amurrado que le quitaron la paletita"
http://hispankillers.foroes.net/t346-nuevo-mapa-stupid-face
"comportandoce como un nino que le quitaron la paletita, Playero, you need to get over it kid, Si te enamorastes de Melina, y ella no te queria dar un pedazo, eso no son maneras de comportarte"
http://elrincondelchisme.blogspot.com/2010/12/como-duele-ser...
In other words, it's a metaphor implying that they've been frustrated in some way; perhaps they've lost something, perhaps they've been preventing from acquiring something; perhaps one of their relatives has been arrested. The point of the expression is that whatever it is, they're in a strop about it, having a tantrum, like a kid.