Several options (and you have likely thought of more)
1) There are no functional changes in our basic reactions to parenting demands; however, greater social networking has put greater restraints on parental action. There are relatively few sanctions for generosity and nurturance -- indeed, there is some evidence that generosity is harder to notice than selfishness, a difference that may rest in our evolved natures. However, we (all of us) are extremely alert for complaints whether from children, our mates, or our neighbors. "What will they think of me if ...?"
2) Children are usually in a selfish mode perhaps even from conception. Their priorities are growth, occupation of territory, and assumption of power. I suspect parents can be usefully viewed as an extended phenotype of the child. Parents will do anything to alleviate complaints, disadvantages, or hardships associated with their child's expectations. My most difficult task is sometimes that of getting parents to ignore their children's whining, blackmail, or other retaliative strategies.
3) Environmental conditions have changed from r-conditions to K-conditions. K-selection implies that the environment is near its carrying capacity, there is a drop in fecundity, and a greater investment in each individual child. There are fewer of them, so each is more "valuable." The contrasting state, r-selection, is associated with many many children and little investment in any of them. It is also associated with relatively abundant resources (although Ed Wilson appears to have a different stance recently on this point and I need to check if he's changed from his initial writings)in a perhaps unstable setting. Thus, the most profitable course is to use up the resources that are available and then adapt to occupy the next available setting.
Many sections of our country have moved into K-selection. There are more political liberals (and advocates of children's "rights") in Cambridge than in Wyoming. Tactics for child rearing will be different simply as a result of population pressure. (Remember as population gets closer, however, to saturating the environment, then various kinds of social disruption become more likely and kids become expendable as adults focus on personal survival.)
In summary, my hunch is that as you move to more rural conditions or have more children, the authoritarian, physical styles of managing children will be more evident.
I personally lament some of the changes I observe in child and adolescent conduct over the past two decades. There is less cooperation between adults in different households in regard to supervising children as a neighborhood project. Indeed, there may be NO neighborhood! Bart is "cool" but also teaches that parents are fools. Beavis and South Park are similar tutors.
Part of having a higher population level but without cohesive neighborhoods is that it becomes possible to prey on other children and adults from adjoining blocks. r-Selection conditions again apply if we consider the opportunity to take advantage of other people as part of the environment. Given the lack of recurring relationships between individuals, there is far less incentive to treat each other fairly. Parents will defend their child from the complaints of a stranger or even a school teacher. Children profit from such defenses but may become more predatory as a result.
Hope these thoughts are helpful ...