Parcellation of the human cerebral cortex into functionally distinct and meaningful regions is important for understanding the human brain. Although there are plenty of studies focusing on functional parcellation for adults, longitudinally-consistent functional parcellation of the rapidly developing infant cerebral cortex at multiple ages is still critically missing for understanding early brain development. Due to the dramatic changes of the cortical structure and function in infants, it is challenging to both capture the meaningful changes of the boundaries of functional regions and keep the parcellation as longitudinally-consistent as possible. To address this problem, we propose a longitudinally-consistent framework to jointly parcellate a population of infant cortical surfaces at multiple ages. Specifically, first, a population-average representation of the functional connectivity profile is constructed at each vertex at each age. Second, the correlation of functional connectivity profiles between any two vertices on the average cortical surfaces is computed. Notably, this correlation computation is performed not only within the same age but also across different ages, weighted based on the age difference, thus forming a large comprehensive similarity matrix. Such similarity measurements encourage to assign similar vertices to the same parcels, even for the vertices on the average cortical surfaces from different ages, and thus hold the longitudinal consistency. Finally, we apply the spectral clustering method on the large similarity matrix to generate an initial joint parcellation for all average surfaces, and further employ a graph cuts method to produce the spatially-smooth longitudinally-consistent parcellations. The proposed method was applied to a longitudinal infant brain MRI dataset to jointly parcellate infant cortical surfaces at 7 different time points in the first 2 years of age. The results show that our parcellations not only capture the evolution of functional boundaries but also preserve the longitudinal consistency.