Former Cosatu general secretary Zwelinzima Vavi and EFF leader Julius Malema are plotting to overthr.
Prime Minister Barnabas Dlamini said the king now owned a McDonnell Douglas DC-9 twin-engine jet, but the plane was a gift to be used by Mswati and his 13 wives.
"His majesty's government has the honour to announce to the nation that the king has received a gift of a DC-9 aircraft from development partners and friends of the king, to be used by their majesties for travels abroad," Dlamini said.
Mswati celebrated his birthday on April 19 with a $386,000 (about R3-million) partly paid for by the state.
He is rated by Forbes magazine as the world's 15th richest monarch with a personal fortune of $100-million (R776-million), but he reins over one of the world's poorest nations.
The Swaziland Diaspora Platform, a human rights group based in South Africa, has rejected the government's explanation for the jet.
"No development partner would want to be anonymous. By their nature, development agencies are transparent.
"We demand the release of full details of the donor and the value of the jet," spokeswoman Ntombenhle Khathwane said.
The banned Pudemo opposition also accused Mswati of using taxpayers' money.
"He was secretly using it and only now do they feel bold enough to admit it. We have long said that this monarchy is a huge drain to the economy," spokesman Zakhele Mabuza said.
Mswati in 2002 secretly used R28-million for a deposit on a 19-seater Bombardier Global Express long-range jet but the move was halted after a huge uproar from parliament, opposition parties and civil society.
Swaziland is currently facing an economic crisis that has sparked protests in the tiny kingdom.
Opposition groups, labour and students have protested to demand more democracy in Africa's last absolute monarchy.