Translation:Their parents educated them in Esperanto.
Yes, it's quite normal to just use "la" where the context makes it fairly clear. Another similar case is with things like "Lavu la manojn!" (literally "Wash the hands!"), where in English I think we would always say "Wash your hands!" but either "la" or "viajn" is fine in Esperanto. And of course, "la" is easier to say and to type. :)
Neither, really. "Eduki" is really very much like the English word "raise" or "to bring up" in spite of the superficial similarity to "educate." So the sentence means that as the children were being brought up and taught all the normal things that good parents teach their children, this was done in Esperanto. As for their education (at school) or formal instruction in Esperanto grammar, this sentence tells us nothing.
I think Robot-Dreams's question is valid too. That's why I answered it.
It might not apply in this case, but it's worth keeping in mind that Wells is a dictionary of British English.
Check out the definitions at the Reta Vortaro. The English definitions I've checked mention school. The RV definition does not. Not only does it say "Direkti la elvolviĝon de fizikaj, moralaj kaj intelektaj kapabloj de infano aŭ junulo" but it also points out that animals and plants can be "educated" as well.
Am I? (That is, am I really acting like you made a pervy comment and am I really being "so patronizing"?) I don't think so. I simply reacted to your comment about the "superreala" experience of meeting a kid who speaks Esperanto (I presume you meant surreal) by pointing out that these are just kids. They are not "kurioziaj skaraboj" -- odd beetles or sideshow freaks to be put in jars and shown around. I absolutely understand your excitement of having had this experience and your desire to share it here - but at the same time, I'm asking you to consider what it would be like if people were saying the same thing about you ... all the time... again and again. Your comment was innocent and understandable. I am simply asking you try to see the unintended consequences of it.