Being a campesino in Europe or in Latin America has no disrespectful meaning, it is simply a statement of fact. Changing socio-economic status in one generation is much harder and unusual in Europe and LA than in North America. It is unusual for country folk (our politically correct term) to have a son become a judge.
It's not insulting, I think it's rather inspiring. I live in India and I've known people, read and heard stories about how; despite having relatively poor economical background; the next generation with their hard work topped all India exams which nearly a million people give. And brought pride to their parents.
"Country folk" is more accurate IMO because el campo means the countryside. And I think the only reason the word "peasant" has a negative image is because when most English speakers hear the word it's context is in the feudal system, where peasants were considered subordinate. We don't see being peasants/country folk as bad (well, unless you're a snob ;} ), it's the word that has negative connotations.
Using the definite article ("the") in the first sentence ("The judge is the farmer's son") implies/requires that we know the specific farmer in question. The meaning shifts from talking about the judges parentage in a general sense (his parents are farmers) to talking about his being the son of a specific farmer.
As far as why we don't say, "the judge is a son of peasants" ... I think it's just idiom.