You are, I believe, correct about parallel construction. It is notably easy for police to find probable cause to effect a search, which is half of what makes parallel construction disquieting. But they do have to actually find some cause to conduct the search. They can't just go with the intel they get from inadmissible evidence; if they do, the new evidence from the unwarranted search is "fruit of the poisonous tree", and can be excluded from trial.