Sarea also claimed they used a new, long-range ballistic missile which they had allegedly manufactured themselves in an attack on Thursday targeting the city of Dammam in Saudi Arabia.
"At least 19 soldiers were killed and others wounded", he noted.
At least 20 soldiers were killed when militants of the Yemen-based Al Qaeda branch launched an armed attack against a base of the newly-recruited security forces in the country's Abyan province on Friday, officials said.
The first attack was a suicide vehicle bombing carried out by extremists on a police station that killed 13 police officers and wounded several others, a security source said.
The United States considers AQAP the network's most risky branch and has waged a long-running drone war against its leaders.
The attack is the second in as many days on a camp for the Yemeni Security Belt forces backed by the United Arab Emirates, a member of the Western-backed Arab coalition battling Houthi rebels who control swathes of the country.
In Thursday's attacks in Aden, the Houthis announced they had fired a ballistic missile at the camp of the UAE-trained forces.
The missile struck about five meters (yards) from the viewing platform and a senior commander was among the dead, a photographer reported.
The simultaneous attacks took place yesterday and the targets were a police station, for which the terrorist organization Islamic State (IS) claimed responsiblity, that left a preliminary death toll of 11; and a military parade, in which 49 soldiers died.
Aden is controlled by Yemen's internationally recognised government and its supporters in the Saudi-led military coalition, which has been fighting the rebels since 2015.
The Yemeni government said Thursday the "source and goal (of the attacks) were the same".
Houthi spokesman Yehia Sarea said in a speech on rebel-run Al-Masirah TV Friday the rebels had gathered "precise intelligence and secured the collaboration of some elements" inside Aden, adding that the attack was part of a pre-emptive strategy aimed at "thwarting plans for escalation by the coalition of aggressors".
The conflict has killed and wounded tens of thousands of people and resulted in the world's worst humanitarian crisis, according to the UN.