The hunger strikers, around 20 people drawn from several opposition parties, said on Thursday the appointment showed Azerbaijan's ruling clan was trying to turn the country into a monarchy.
The country's 80-year-old President Heidar Aliyev made his 41-year-old son the new prime minister earlier this week.
The move was widely seen as a precursor to Ilham Aliyev succeeding his father as head of state.
"We regard this as an attempt to usurp power in this country," said Iqbal Agazade, a deputy in parliament who is leading the protest.
"We believe that Ilham Aliyev should not be prime minister and should not run the country."
However, Aliyev junior dismissed the hunger strike.
"The fact that these people are on hunger strike makes me laugh. I know that in the evening they sneak out to eat kebabs," he said.
President Aliyev, who is 80 and has been in a Turkish clinic for a month, was flown to a clinic in America on Wednesday.
Ilham Aliyev said his father's health problems were caused by returning to work too soon after collapsing on live television on April 21.
Haidar Aliyev is said to be
"He returned to work before his treatment had been completed, in defiance of doctors' orders. Now he feels well but he still needs further treatment," he said.
There has been media speculation that Aliyev travelled to the Ohio clinic for heart surgery.
The clinic has a world-renowned reputation in this field and Aliyev had a heart bypass operation there in 1999.
But his son said: "I think that (an operation) is not needed. Should such a need arise, an operation will be performed.
"There are highly-qualified doctors there. But no such need has emerged so far."
Heidar Aliyev, a former KGB general and member of the Soviet Union's Politburo, has run Azerbaijan for most of the past three decades.
He is the world's oldest democratically-elected president.
Both he and his 41-year-old son are registered as candidates for presidential elections set to take place on October 15.
But it is widely expected that Aliyev senior will withdraw his candidacy, leaving his son to take a shot at the presidential office.