The couple were at a petrol station in the small town of Barrancas on Saturday when they were abducted by armed men on motorcycles. Diaz's mother, Cilenis Marulanda, was rescued within hours by police that set up roadblocks around the town of 40,000 people, which is near Colombia's border with Venezuela.
But his father, Luis Manuel Diaz, remained missing. On Monday, special forces searched a mountain range that straddles both countries and is covered by cloud forest. Police also offered a £40,000 reward for information leading to Diaz's father.
Officials said they did not rule out the possibility that the elder Diaz could have been smuggled into Venezuela, where he would be beyond the reach of Colombian police.
Diaz is a mainstay in the Colombia national team and for Liverpool in the Premier League, which he joined last year in a deal worth £37million.
The 26-year-old forward was absent from Liverpool's match against Nottingham Forest on Sunday. His teammates expressed their solidarity with the Colombian by holding up one of his shirts after scoring the team's first goal in their 3-0 victory.
The abduction of Diaz's parents comes as kidnappings for ransom and extortion of businesses increase in Colombia despite efforts by the nation's first left-wing government to broker ceasefires with rebel groups. Criminals and rebel groups in the country have long kidnapped civilians for ransom in order to finance their operations.
So far, none of the armed groups operating in Colombia have claimed responsibility for kidnapping Diaz's parents.
