The healthy development of physical quality in the growth process of young children is an important prerequisite for their happy learning, and the effective teaching of gymnastics for young children can realize the effective improvement of young children’s physical quality and various functions. This paper constructs a hierarchical structure model to assess the teaching ability of early childhood gymnastics using the hierarchical analysis method. In order to effectively analyze the degree of influence of each index in the hierarchical structure of gymnastics teaching ability of young children, the quantile regression model under the parametric model is used for verification, and the model parameters are estimated by penalized regression and two-stage regression. Taking the teachers of ten public kindergartens in S city as an example, the relevant influencing factors of the hierarchical structure model of early childhood gymnastics teaching competence were analyzed in depth. The results show that under the τ = 0.5 quartile, instructional design ability has the most significant effect on the hierarchical structure of young children’s gymnastics teaching ability, and its coefficient value fluctuates around 0.45, whereas the degree of influence of teaching research ability teaches and the coefficient estimate fluctuates between [0,0.15]. The ability to teach gymnastics to young children is related to the healthy development of young children’s physical fitness, and early childhood teachers need to strengthen their teaching-related skills with a view to better ensuring the happy growth of young children.