The Syrian Observatory for Human Rights said a number of people were injured, some seriously, in the attacks on the Sukkari and Ashrafiyeh neighbourhoods and that the number of dead was expected to rise.
The monitoring group distributed gruesome images from the scene of the attacks on Sukkari, where 25 people were killed, showing residents holding body parts of victims from the raid.
Observatory director Rami Abdel Rahman said a first attack created chaos, with residents and medics rushing to the scene to help.
Another six people died in a separate barrel bomb attack in a rebel-held area of the Ashrafiyeh neighbourhood, in the north of the city.
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Syria's regime has waged a fierce aerial offensive against rebel-held parts of Aleppo since last December.
The Observatory said in May the campaign had killed nearly 2,000 people -- more than a quarter of them children -- since the beginning of 2014.
Rights groups have decried the regime's use of barrel bombs as unlawful because they lack an aiming mechanism, causing indiscriminate casualties.
Elsewhere, the Observatory said 13 people, including eight children, were killed on Sunday night by rebel shelling of a government-held area in western Idlib province.
More than 162,000 people have been killed in Syria's conflict, which began with peaceful protests against President Bashar al-Assad's regime in March 2011 before spiralling into a bloody war after a massive government crackdown on dissent.