These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
YMMV: The Neanderthal Parallax
Mary Suetopia: The Neanderthal world has very little crime, hasn't had a war in hundreds of years, and lives in harmony with nature.
Deconstructed in later books. The Neanderthal society deliberately chose to breed violence out of its population because due to the size and strength of an average individual, a minor dispute could easily end with someone getting killed. One such scene is depicted in the first book; a dispute that two characters escalates into one having his mandible essentially destroyed with one punch-it's later stated that most past assassinations consisted of someone simply walking up to the target and caving their head in by their fists alone. The problem arose when they started sterilizing anyone sharing half or more of the violent person's genome as well. It's implied that the sterilization policy has actually created a pretty significant problem with domestic violence, since the victim can't report the crime without condemning any children they might have to the same fate as the abuser.
Sawyer has argued both in other novels and Real Life for something like the surveillance system of the series, feeling privacy is not only overrated but dangerous, and increasingly disappearing anyway.
Their society isn't entirely perfect. At least one major hole in their criminal justice system is explored in detail in Hybrids, and individuals predisposed to violence and antisocial behavior continue to appear in the Neanderthal population despite their eugenics program. As for voting, the only limitation is age; otherwise anyone who reaches a certain age can vote. The Companions are limited in not being able to report back to the central repository if one is too far away from a city (which would make H. sapiens space technology all that more valuable to the Neanderthals—a satellite monitoring system would improve the efficiency of the Companions dramatically). And Ponter was absolutely crazy about consuming domesticated chickens dipped in a flour-based batter seasoned with 11 agricultural products and fried in vegetable oil, none of which would have been possible in his world. Overall the two sides seem about equally balanced.
I doubt they can be considered equally balanced, if, as the book claims, they have no war, far less crime, sexual liberation and lack religion (obviously YMMV will vary about those last two).
They're not "more sexually liberated"-if anything they're less. The government keeps men and women segregated for most of the month, and only allows them to have children once every ten years.
True, but they all practice bisexuality (though individuals with solely heterosexual or homosexual attraction, not to mention anything else, are apparently unknown). Ponter also mentions that adultery is not considered a big deal, as their superior sense of smell lets them tell instantly if a partner has slept with someone else, or a woman's gotten pregnant, thus letting men avoid being cuckolded (the fear of which is something anthropologists have postulated as a major factor behind possessive monogamous practices). When the sexes meet, it's also apparently acceptable for them to sleep with people beside their primary partners, given Adikor's causal question of whether Ponter had sex with a woman he isn't mated to (and one who had sought to get Adikor castrated for supposedly murdering Ponter in the previous novel, no less).
The only real difference between Neanderthal voting practices and ours is degree. We allow people to vote on any issue as soon as they reach legal adulthood, no matter how little knowledge and experience they have, not to mention the fact that the brain is still in adolescence; the Neanderthals only allow those with (what they consider) a reasonable amount of life experience to vote, but even they can vote on any issue, informed or otherwise. It also seems that Neanderthalls aren't allowed to become members of their legislative bodies before they reach even greater age, as these are called Gray Councils in reference to their hair color. "Gray" is a synonymous for wisdom among Neanderthals-they appear to have gerontocracy, or rule by the elderly.