No, if a warrant is obtained using illegal evidence, then you poison the warrant and all the evidence arising from it.
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).
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).