Is it permissible to offer one continuous charity (sadaqah jariyah) jointly for both deceased parents, or must there be a separate continuous charity for each of them?

We understand from your question that you wish to dedicate the entire charity to your parents, which is permissible and there is no harm in it, and it will be an ongoing charity (Sadaqah Jariyah) for them. However, it is better for each of them to have a complete charity dedicated to them individually. Sharing the reward does not mean complete fulfillment or receiving the same reward as someone who performed the sacrifice individually. Therefore, strive to dedicate a specific ongoing charity to each of them. As for participating in building a mosque, there is no harm in dividing the charity between them, so that each of them would, for example, contribute half of the mosque. Building mosques and assisting in their completion is considered an ongoing charity.