This seems like a rather vague claim
Including the two neighboring counties, out of nearly 750,000 criminal cases during that time, only about 3 or 4 dozen involved parallel construction, and most of those were gang cases in which the parallel construction involved one of the gang members turning on his homies.
It's big news in local legal circles when the prosecution tries to use parallel construction to get evidence into the record because it happens so rarely.
It does happen more frequently at the federal level, but they also have significantly more resources to conduct investigations along parallel paths.
I believe it depends on context. If you're in police custody and being questioned, you need to be read your Miranda rights. If the police show up to your door and you suddenly blurt out a confession, I don't think it matters.
However, once the police start asking questions, they have to show they had not created a "custodial" situation in which the suspect was not free to leave (or in this case, to close the door and tell the police to fuck off). In most states, the burden is on the defendant to show that a custodial situation was created; in some states, that is presumed and the police have to show that a custodial situation was not created.
For the record--I am former public defender. And I successfully used the failure to properly Mirandize a client successfully many times to block the prosecution from introducing evidence.
Parallel construction happens way less frequently than people think it does, since it requires that there be an actual parallel path to the evidence that would have been available at the time the evidence was improperly obtained. It's basically only upheld when evidence was collected improperly through one means (i.e., a confession without Miranda disclosure), but could have been collected properly through other means available at the time of the improper collection (i.e., if the contents of the confession would have been revealed by a standard CSI search).