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Since the asteroseismic revolution, availability of efficient and reliable methods to extract stellar-oscillation mode parameters has been one of the keystone of modern stellar physics. In the helio- and asteroseismology fields, these methods are usually referred as peakbagging. We introduce in this paper the apollinaire module, a new Python 3 open-source Markov Chains Monte Carlo (MCMC) framework dedicated to peakbagging. The theoretical framework necessary to understand MCMC peakbagging methods for disk-integrated helio- and asteroseismic observations are extensively described. In particular, we present the models that are used to estimate the posterior probability function in a peakbagging framework. A description of the apollinaire module is then provided. We explain how the module enables stellar background, p-mode global pattern and individual-mode parameters extraction. By taking into account instrumental specificities, stellar inclination angle, rotational splittings, and asymmetries, the module allows fitting a large variety of p-mode models suited for solar as well as stellar data analysis with different instruments. After having been validated through a Monte Carlo fitting trial on synthetic data, the module is benchmarked by comparing its outputs with results obtained with other peakbagging codes. An analysis of the PSD of 89 one-year subseries of GOLF observations is performed. Six stars are also selected from the Kepler LEGACY sample in order to demonstrate the code abilities on asteroseismic data. The parameters we extract with apollinaire are in good agreement with those presented in the literature and demonstrate the precision and reliability of the module.