I would want to argue that there are actually two levels of conflict that occur in this excellent story. There is of course the overt external conflict that occurs between George and Lydia and their children, Peter and Wendy. This of course reaches its climax when George insists on shutting down the nursery and preventing the children from accessing it. Consider what Peter says to his father as a result and how he behaves towards him:
"Don't let them do it!" wailed Peter at the ceiling, as if he was talking to the house, the nursery. "Don't let Father kill everything." He turned to his father. "Oh, I hate you!"
Clearly there is a tremendous conflict occurring between the children and their parents over the children's apparent obsession with the nursery. However, at the same time, this quote points towards a much deeper and more important conflict concerning the nature of reality. Brabdury presents us with a future world where technology has become so sophisticated that what is unreal and illusory is becoming more real than reality itself. Note the way that Peter in the above quote addresses the nursery as if it were a sentient being, and talks of the possibility of the images that the nursery creates being "killed." Note what David McClean says to George about the importance of the nursery to his children:
This room is their mother and father, far more important in their lives than their real parents.
Thus, far more fundamental than the conflict between the children and the parents is the deeper conflict which technology has created: the way that what is unreal and illusory supplants what is real.
While the quick answer to the question of conflict Ray Bradbury's short story, "The Veldt," may be the antagonism of the children, Peter and Wendy [ironically, names from Peter Pan], towards their parents, the main conflict is that of the dominance of the house over the family. For throughout the narrative there are allusions to the overwhelming control that the house exerts over its occupants. In the first main paragraph, for instance, there is a foreshadowing of the problem as George's wife
...paused in the middle of the kitchen and watched the stove busy humming to itself, making supper for four....
When George and his wife express their concern about how the children "live for the nursery," George suggests that it be locked, and his wife comments,
"I don't know--I don't know," she said blowing her nose, sitting down in a chair that immediately began to rock and comfort her. "Maybe I don't have enough to do. Maybe I have time to think too much. Why don't we shut the whole house off for a few days and take a vacation?...
"You look as if you didn't know what to do with yourself in this house, either. You smoke a little more every morning and drink a little more every afternoon and need a little more sedative every night. You're beginning to feel unnecessary, too."
Clearly, the house has taken dominance in the famil. George and Lydia eat alone at dinner while the children enjoy themselves at a plastic carnival. And, when George questions them about the African veldtland, they look confused. So, George hurries to the nursery where he finds no veldtland, but instead there is Rima in a lovely forest.
The next day Peter asks his father if he is going to lock up the nursery for good. When George tells him "It all depends," Peter coldly replies that he would not want the nursery locked. He is also adverse to having to brush his teeth himself and tie his own shoes.
Growing increasingly concerned the parents call a child psychologist to come to the house; there, he counsels George and Lydia that they have
let this room and this house replace you and your wife in your children's affections. This room is their mother and father, far more important in their lives than their real parents.
This, then, is the main conflict of Bradbury's story. not the conflict of the children with their parents since the quarrels of Peter and Wendy with their parents are not interpersonal, but are, instead arguments about the house. More specifically, the children are infatuated with the house and its creative powers:
"Don't let them do it!" wailed Peter at the ceiling, as if he was talking to the house, the nursery. "Don't let Father kill everything." He turned to his father, "Oh, I hate you!....I wish you were dead!"
"We were, for a long while. Now we're going to really start living. Instead of being handled and massaged, we're going to live."
These, however, are ironic words, for the house wins the conflict: The parents die in the Veldt. So, while there is a parent/child conflict; the main conflict is that of the parents/veldt in which the house replaces the parents as those in charge of both the house and the children. Perhaps, then, the main conflict may be termed Man vs. Technology because whoever controls the machine will have power in the house. But, the machine wins by killing the parents and by destroying the humanity of the children.