Typically yes, but more specifically it depends. Consider the following parallax example, which goes directly to your question:
Hold an extended finger several inches from your face, close one eye and rotate your head about a vertical axis (yaw) as you would were you to turn your head to look left/right. You will see your (foreground) finger traverse across the background. Parallax occurs here because the lens (your eye) entrance pupil is not located on the rotation axis (your neck), but instead some horizontal distance from it.
To eliminate parallax the lens entrance pupil must be located on the axis about which the lens will be rotated. From this it follows (for a levelled panoramic head) that:
- For a vertical-only image set: The lens entrance pupil need only be located correctly relative to the 'camera arm' the body connects to (rotation axis is horizontal, through the same arm at its other end)
- For a horizontal-only image set: The lens entrance pupil need only be located correctly relative to:
- The horizontal arm if all you have is a rail, or;
- The horizontal and camera arm if using a panoramic head because both affect the entrance pupil location relative to a vertical rotation axis (through the panoramic head/tripod)
- For a vertical and horizontal image set both preceding bullet points apply, but the second incorporates the first, hence governs.
Yes, normal practise is to level the head, then rotate the horizontal and camera arm to capture horizontal and vertical images respectively.
Yes, as the lens entrance pupil is still located:
- On the relevant rotation axis in the case of a (tilted) vertical or horizontal image set, and:
- At the intersection of both rotation axes in the case of a (tilted) vertical and horizontal image set.
However, with the camera tilted so too will be the image set, traversing diagonally across the field of view. This would make no difference in the case of a 360x180 (spherical) panorama (albeit is a non-standard means by which to shoot it), but for coverage less than it you potentially would require to shoot more images in the set to ensure the desired composition was captured.