Are non-profit entities allowed to host TeamSpeak 2 and TeamSpeak 3 at the same time?

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PLEASE NOTE: NPLs ARE NO LONGER AVAILABLE

However, you can apply for TeamSpeak Sponsorship.  Please visit: teamspeak.com/sponsorship for more details. You can also purchase a Gamer License right here: https://www.teamspeak.com/features/pricing/

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Yes they can.

Each product has its own restrictions and licensing requirements. Non-profit entities can host a TeamSpeak 2 server with up to 2 virtual servers and 1000 slots in addition to a TeamSpeak 3 server with up to 1 virtual server and 32 slots (unregistered) or with up to 2 virtual servers and 512 slots (registered).