The recent developed cognitive radars can implement flexible work modes with programmable modulation types and optimized modulating values for each mode definition parameter. Automatic analysis of these work modes is a significant challenge for modern electromagnetic reconnaissance receivers. In this paper, a Multi-Output Multi-Structure (MOMS) learning-based processing framework is proposed for Joint inter-pulse automatic Modulation Recognition and Parameter Estimation (JMRPE-MOMS). We propose a label construction method as a feature interpretation method of the network to facilitate MOMS learning and utilize the correlations between labels for performance gain. Moreover, an LSTM-Transformer is designed to mine deep time-series characteristics, which can model local and global relationships and reduce quantization loss. The proposed framework can perform joint modulation recognition and parameter estimation (JMRPE) tasks simultaneously with flexible output structures including scalar output and vector output with fixed or variable sizes. Extensive simulations are performed based on the simulated radar work modes defined with pulse repetition interval (PRI) sequences. The simulation results validate the effectiveness and superiority of the proposed method especially under non-ideal electromagnetic environments.