I think these theories are still greatly applicable to understanding how learning occurs. For example, parents who know Piaget's theory of the sensorimotor stage will likely give their young children lots of exposure to various colors, sounds, and textures. They engage their children in games like peek-a-boo to help them realize the concept of object permanence. They play simple games, such as stacking blocks, with these young learners, which helps them understand that their actions have direct consequences on the world around them. These types of parenting strategies are frequently recommended by pediatricians, so Piaget's theories are still referenced for this age group, even if people don't attribute these ideas to him by name.
Adults who work with preschoolers and young, elementary-aged children (the preoperational phase) know that they are very concrete in their thinking, especially in those earlier years. Therefore, they don't innately understand the perspective of others, so they don't empathize well—and shouldn't be expected to, at first. This sense of a concrete world also prohibits their ability to read, as they have to first learn that symbols can represent ideas and objects. And all of them don't transition out of a concrete understanding of the world at the same time, which explains to some degree why children learn to read at different ages.
We see evidence of Piaget's theories of formal operational thought in the way we design curricula in schools. Since this type of abstract thought doesn't develop until around age twelve, we don't typically ask students younger than that to examine complex philosophies or political issues.
Piaget believed that children do not simply gain more information as they get older; instead, they are able to think about information in increasingly complex ways. This is reflected in the types of questions teachers ask of children of various ages and the tasks they ask their students to complete. The idea that children must investigate and experiment in order to truly learn new concepts is a great part of the "hands-on" and kinesthetic movement of educational philosophy that still exists today.
In short, threads of Piaget's theories are still greatly utilized in educating children of all ages.