Pashinyan's party won 71 seats in the June 20 election, while 29 went to a bloc headed by former President Robert Kocharyan.
Another quarantine destination is Armenia, which is now in conflict with Azerbaijan and has no restrictions for travellers from India.
Azerbaijan said Armenian forces opened fire at its positions on the Kalbajar section of the border.
Moscow, which has a military base in Armenia, has deployed some 2,000 peacekeepers in and around Karabakh to oversee the ceasefire.
The Armenia defence ministry said an intense shootout took place near the village of Sotk at the north-eastern sector of the border shared by Azerbaijan.
Under Armenian law, snap elections can be held after a premier resigns and the parliament fails twice to choose a new one.
Armenian acting Vice Premier Armen Gevorkyan's apparent move to secure support from the Kremlin came as protesters in the capital Yerevan took to the streets for a fresh day of rallies.
The country's veteran leader Serzh Sarkisian stood down on Monday from his new post as prime minister after days of protests by demonstrators who accused him of a blatant power grab.
According to Armenian law, lawmakers in the parliament, where Sarkisian's Republican Party have a majority, have a week to propose new candidates for PM and could hold a vote on May 2.
Residents of the capital, Yerevan, poured out on the streets to celebrate his stunning departure. People hugged and kissed each other, and motorists honked their horns.
Sarkisian, who served as president for a decade, was last week elected as prime minister with sweeping powers, triggering the protests.
Thousands of people in Armenia have been protesting for 10 days against ex-president Serzh Sarkisian taking up a new position as premier with sweeping powers.
With protests against Armenian PM Serzh Sarkisian escalating, here is all you need to know about the country's troubled history
Armenia is facing a deepening political crisis, after talks between Prime Minister Serzh Sarkisian and protest leader Nikol Pashinyan failed.
Armenian police said protest leader Nikol Pashinyan was forcibly taken from a protest rally, dismissing reports of his arrest as riot police and demonstrators clashed in Yerevan.
Armenia's political turmoil deepened as Prime Minister Serzh Sarkisian stormed out of talks with opposition leader Nikol Pashinyan, who was later forcibly removed from a protest.