The message from Ali Larijani less than a week after talks resumed appears aimed at both envoys from the West and Iran's negotiation team, which is led by Foreign Minister Mohammad Javad Zarif.
It also highlights the political jockeying inside Iran between backers of moderate-leaning President Hassan Rouhani and hard-liners wary of his outreach to Washington.
Larijani's comments follow appeals by some members of the US Congress to tighten sanctions on Iran despite the nuclear negotiations and historic diplomatic breakthroughs last month, including President Barack Obama's telephone chat with Rouhani.
Iran's ruling clerics approve all major policies and decisions, but parliament holds enough clout to potentially disrupt talks in response to Western demands to curb the program.
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Details from last week's talks have remained tightly guarded, but short-range priorities have been made clear. The US and allies seek to roll back Iran's highest-level uranium enrichment, which is several steps away from weapons grade. Iran wants the West to start withdrawing sanctions, which have hit Iran's vital oil exports.