Fariñas is a Cuban doctor of psychology, independent journalist and political dissident. He has staged 23 hunger strikes over the years to protest at various aspects of the Cuban regime.
In his latest protest with other Cuban human rights activists, who are also on hunger strike, Fariñas has not eaten or drunk anything since 19 July.
Fariñas is a former winner of Parliament's Sakharov prize, and has recently stated that he is "ready to die in the struggle against censorship in Cuba."
The 54-year-old is protesting against the alleged torture and mistreatment of political prisoners by the Cuban government.
Fresh concern about his plight has been voiced by the Chair of Parliament's human rights subcommittee, Spanish Socialist deputy Elena Valenciano.
Speaking on behalf of the subcommittee, Valenciano said: "We are extremely concerned about the health of Fariñas. I call on the Cuban government to ensure the integrity of his health and to address his call for immediate improvement of the treatment of the political prisoners in Cuba."
She added, "We call on the Cuban government to abide by its international commitments stemming from the UN convention against torture and other cruel, inhuman or degrading treatment or punishment.
"The subcommittee will closely follow this case and the respect for human rights and democracy in Cuba, this being one of the main cornerstones of the EU-Cuba political dialogue and cooperation agreement," she said.
Fariñas was awarded the Sakharov prize for freedom of thought in 2010.
In presenting the award, Parliament commended Fariñas, saying that he was a "symbol of the fight for freedom of speech".
It is the third time that the award has been presented to Cuban dissidents.
In December 2010, the Cuban government denied Fariñas an exit visa necessary to travel to Strasbourg to accept the award. In response, Parliament said that it would have an empty chair to represent him at the ceremony.
At the time, Fariñas said, "I believe that the Cuban government has shown over the years that it is behaving in an arrogant manner."
He is currently being treated for the ill effects of his continuing hunger and thirst strike and is calling on Cuban President Raul Castro, younger brother of Fidel Castro, "to halt mistreatment of political prisoners in Cuba."
Currently 21 human rights activists are on hunger strike to protest against alleged police violence in Cuba.
A Cuban government spokesman hit back, saying, "Cuba will not accept pressure or blackmail, important western media groups are again calling attention to a prefabricated lie. The consequences (of the hunger strike) will be their responsibility, and theirs alone."