"Don't we already allow anonymous tips? "
We allow anonymous tips that can get police's attention on something. Our rights then apply to restrict their behavior to what's sensible. Parallel construction is them violating our rights, producing (forging?) evidence, lying about the methods that produced it, then faking/finding a new route to convict us, and then we're convicted on weight of that and/or secret evidence. This poses extra risk due to (a) damage from parallel construction activities themselves, (b) ability to target dissenting/unpopular groups as FBI has long history of doing with permissive powers, and (c) inability to challenge the quality or effectiveness of their selection & evidence gathering process vs more open processes.
" The only thing I am addressing is how society should treat the discovery of random bits of information put into the public domain by thieves."
What's you've done is force the situation to be an extremely narrow case that doesn't represent how parallel construction is used at all. It's normally LEO's focusing on specific groups with surveillance, subversion, and informants. What they focus on vs what harms society most has a disconnect that reveals more political than protective reasons. One of reasons we have accountability measures in the first place. Analysis stemming from thieves breaking into a house & dumping evidence is a very, uncommon case.
Let's look at it anyway. Regular process usually involves searches that you would know about so you can make sure nobody plants evidence. Thief breaks into your house, leaves child porn there, delivers anonymous tip to authorities that you possess child porn, they search you, they find it, and you're convicted. Still trust public safety via thieves method?
Alternatively, a politician wants to be insanely powerful. All he has is parallel construction, police authority, and some surveillance tech. Starts using it selectively on key members of Congress, judges, media people, and so on. Finds dirt on them, but doesn't act on it: extorts them for appropriations, expanded legal powers, hushing on bad stories, and so on. Might be able to pull this stuff off for a long time with untold damage. Wait, why speculate: I'm talking about how J Edgar Hoover amassed power for him and his FBI. Process could be repeating with NSA's surveillance power & Congress constantly ignoring their corruption but we won't know because all of that is secret. ;)
Shifting power to executive branch for secret accusations against people with less and less protections only results in predatory behavior. Both U.S. and foreign history are pretty clear that this is what usually happens. So, it shouldn't be allowed. Our prior system worked well enough. Too well given number of wrongful convictions and false confessions. Reforming it is a better idea.