Why would he switch from Parseltongue to English and back to Parseltongue?

If Voldemort can lie in Parseltongue, then it's a ruse to conceal that fact. This has the dual effect of causing Harry to accept the literal truth of what VM says in Parseltongue without question, as well as persuading Harry to not even attempt to tell any direct lies to VM in Parseltongue. Those are both strong advantages manufactured from thin air, and VM is clever enough to spin such a deception.

And "snakes can't lie", really? That's a pretty odd inversion of colloquialisms like "speaking with a forked tongue." Ssuddenly I feel like ssomeone is trying to ssell me a bridge.

Inability to lie in Parseltongue has been established clearly enough that it would be very odd for it to be broken now.

Has it? Harry performed one simple test at a time when Voldemort expected that test to be performed, and we don't know that the results weren't influenced by other magic. We haven't seen it tested with occlumency. We certainly haven't seen it tested by a perfect occlumens. Even if we had seen those results it wouldn't be conclusive about what the most powerful wizard alive, who also knows the secrets of Salazar, is capable of.

I think for meta reasons that I'm probably wrong, but based only on in-world evidence I don't see any reason at all to take Voldemort's word that Parseltongue is a language of truth.