Supervised learning methods have shown effectiveness in estimating spatial acoustic parameters such as time difference of arrival, direct-to-reverberant ratio and reverberation time. However, they still suffer from the simulation-to-reality generalization problem due to the mismatch between simulated and real-world acoustic characteristics and the deficiency of annotated real-world data. To this end, this work proposes a self-supervised method that takes full advantage of unlabeled data for spatial acoustic parameter estimation. First, a new pretext task, i.e. cross-channel signal reconstruction (CCSR), is designed to learn a universal spatial acoustic representation from unlabeled multi-channel microphone signals. We mask partial signals of one channel and ask the model to reconstruct them, which makes it possible to learn spatial acoustic information from unmasked signals and extract source information from the other microphone channel. An encoder-decoder structure is used to disentangle the two kinds of information. By fine-tuning the pre-trained spatial encoder with a small annotated dataset, this encoder can be used to estimate spatial acoustic parameters. Second, a novel multi-channel audio Conformer (MC-Conformer) is adopted as the encoder model architecture, which is suitable for both the pretext and downstream tasks. It is carefully designed to be able to capture the local and global characteristics of spatial acoustics exhibited in the time-frequency domain. Experimental results of five acoustic parameter estimation tasks on both simulated and real-world data show the effectiveness of the proposed method. To the best of our knowledge, this is the first self-supervised learning method in the field of spatial acoustic representation learning and multi-channel audio signal processing.