Vaši práci nejsou?
Missing the "to" would just translate to "your birds are not" which wouldn't make sense by itself
This looks like the negative version of the familiar "to je/to jsou" construction, so I don't think you can leave out the "to." "Oni nejsou..." might be an option, but a Czech expert would be better to weigh in on that.
"to" is required in this sentence.
Actually, "Oni nejsou vaši ptáci." is fine, if somewhat unusual. But it would be saying that about some people.
Isn't there a rule about "to" always being second in a sentence?
The second position is not necessarily the second word in the sentence. It's the second 'idea'. Here, "your birds" is the first idea.
In second position, but "vaši ptáci" is in the first position, you can't split it - if you meant that. It is really better to ask directly.
Oh, okay. I didn't really mean that, I was just wondering why it wasn't in second position.
Well, it is.
Hold on. I could say "TO nejsou vaši/tvoji/tví ptáci", couln't I? So why are we even talking about the second place rule here?
To is in the second position here. It is not required to be there. "To vaši ptáci nejsou." is completely fine.