The Britain-based Syrian Observatory for Human Rights, which relies on a network of activists on the ground, said at least 40 people were killed and around 90 wounded in the four blasts, most of them members of the regime forces.
It said mortars also targeted the nearby political security department around the same time of the bombings.
The Syrian security official, however, said most of Wednesday's casualties were civilians.
"We condemn these crimes and this terrorist explosion and we also condemn the countries that conspire against Syria and stand behind the terrorists," said the speaker of the Syrian parliament, Mohammad Jihad al-Lahham, told the assembly Wednesday.
During the course of the 18-month-uprising against Assad, suicide and car bombings targeting security agencies and soldiers have become common in Syria, particularly in the capital, Damascus.
But Aleppo has been spared from such bombings and from the mayhem that struck other Syrian cities, particularly in the first year of the revolt. Then, in February, two suicide car bombers hit security compounds in Aleppo's industrial center, killing 28 people.
The uprising against Assad erupted in March 2011 and gradually morphed into a bloody civil war. The conflict has killed more than 30,000 people, activists say, and has devastated entire neighborhoods in Syria's main cities, including Aleppo.
The city, one of the world's oldest continuously inhabited cities, has been the site of fierce battles for more than two months between regime troops and rebels fighters that have brought relentless shelling and gun battles.
Over the weekend, a fire sparked by fighting tore through the city's centuries-old covered market in the Old City, burning over 500 shops. At 12 kilometers (7.5 miles), it is the Middle East's longest souk and is part of Aleppo's old center that was added in 1986 to UNESCO's list of World Heritage sites.