Benefiting from exploiting the data topological structure, graph convolutional network (GCN) has made considerable improvements in processing clustering tasks. The performance of GCN significantly relies on the quality of the pretrained graph, while the graph structures are often corrupted by noise or outliers. To overcome this problem, we replace the pre-trained and fixed graph in GCN by the adaptive graph learned from the data. In this article, we propose a novel end-to-end parallelly adaptive graph convolutional clustering (AGCC) model with two pathway networks. In the first pathway, an adaptive graph convolutional (AGC) module alternatively updates the graph structure and the data representation layer by layer. The updated graph can better reflect the data relationship than the fixed graph. In the second pathway, the auto-encoder (AE) module aims to extract the latent data features. To effectively connect the AGC and AE modules, we creatively propose an attention-mechanism-based fusion (AMF) module to weight and fuse the data representations of the two modules, and transfer them to the AGC module. This simultaneously avoids the over-smoothing problem of GCN. Experimental results on six public datasets show that the effectiveness of the proposed AGCC compared with multiple state-of-the-art deep clustering methods. The code is available at https://github.com/HeXiax/AGCC.