Yes, this absolutely does not shield YouTube from liability from third parties, since the copyright holder of third-party content included in the video is not a party to the agreement. That's why they have a copyright notice and takedown procedure in the first place, and also the reason for numerous lawsuits against YouTube in the past, some of which they have lost.

To date, many Creative Commons licenses do in fact amount to "permission from that party", but if they start using DRM, those licenses would cease to grant YouTube permission.