The unprecedented possibility to acquire high resolution Satellite Image Time Series (SITS) data is opening new opportunities to monitor the different aspects of the Earth Surface but, at the same time, it is raising up new challenges in term of suitable methods to analyze and exploit such huge amount of rich image data. One of the main tasks associated to SITS data analysis is related to land cover mapping. Due to operational constraints, the collected label information is often limited in volume and obtained at coarse granularity level carrying out inexact and weak knowledge that can affect the whole process. To cope with such issues, in the context of object-based SITS land cover mapping, we propose a new deep learning framework, named TASSEL (aTtentive weAkly Supervised Satellite image time sEries cLassifier), to deal with the weak supervision provided by the coarse granularity labels. Our framework exploits the multifaceted information conveyed by the object-based representation considering object components instead of aggregated object statistics. Furthermore, our framework also produces an additional outcome that supports the model interpretability. Quantitative and qualitative experimental evaluations are carried out on two real-world scenarios. Results indicate that not only TASSEL outperforms the competing approaches in terms of predictive performances, but it also produces valuable extra information that can be practically exploited to interpret model decisions.