There's no reason to assume that the statement is true, it's coming from the police, they could easily be lying.
I'd understand the argument for why they're incentivized to get a warrant in the first place (even assuming parallel construction is a viable alternative to getting a warrant as they may be caught) but that doesn't mean they wouldn't lie after they got caught screwing up.
I'm happy to concede that policing in America has long eroded that trust but it's still a requirement that you obey and openly communicate with anyone identifying themselves as law enforcement.
There is no requirement that you communicate openly with police. See: the Fifth Amendment to the US Constitution.