Enzymatic reaction kinetics are central in analyzing enzymatic reaction mechanisms and target-enzyme optimization, and thus in biomanufacturing and other industries. The enzyme turnover number (kcat) and Michaelis constant (Km), key kinetic parameters for measuring enzyme catalytic efficiency, are crucial for analyzing enzymatic reaction mechanisms and the directed evolution of target enzymes. Experimental determination of kcat and Km is costly in terms of time, labor, and cost. To consider the intrinsic connection between kcat and Km and further improve the prediction performance, we propose a universal pretrained multitask deep learning model, MPEK, to predict these parameters simultaneously while considering pH, temperature, and organismal information. Through testing on the same kcat and Km test datasets, MPEK demonstrated superior prediction performance over the previous models. Specifically, MPEK achieved the Pearson coefficient of 0.808 for predicting kcat, improving ca. 14.6% and 7.6% compared to the DLKcat and UniKP models, and it achieved the Pearson coefficient of 0.777 for predicting Km, improving ca. 34.9% and 53.3% compared to the Kroll_model and UniKP models. More importantly, MPEK was able to reveal enzyme promiscuity and was sensitive to slight changes in the mutant enzyme sequence. In addition, in three case studies, it was shown that MPEK has the potential for assisted enzyme mining and directed evolution. To facilitate in silico evaluation of enzyme catalytic efficiency, we have established a web server implementing this model, which can be accessed at http://mathtc.nscc-tj.cn/mpek.