Four people were injured.
National Police Chief Tito Karnavian said the father exploded a auto bomb, two sons ages 16 and 18 used a motorcycle for their attack, and the mother was with her two young daughters for her attack.
The national police chief, Tito Karnavian, said the family had been in Syria, where the Islamic State group until recently controlled a large swath of territory.
The so called Islamic State group claims responsibility.
East Java police spokesman Senior Commissioner Frans Barung Mangera said 13 people were confirmed dead and 41 who were injured were being treated at several hospitals in Surabaya as of Sunday evening. A new threat has emerged in recent years, inspired by IS attacks overseas. The police have pointed to a marriage that had returned from Syria with ir four children, aged between nine and eighteen years.
Karnavian named the father as Dita Futrianto and said he was head of a cell of Jemaah Anshorut Daulah, an Indonesian militant network affiliated with Islamic State. He identified the mother as Puji Kuswati.
The father drove a bomb-laden auto into the grounds of Surabaya Centre Pentecostal Church.
Suicide bombers who committed terrorist attacks in churches in Indonesia were members of the same family. Based on their remains, Karnavian said the mother and daughters were all wearing explosives around their waists.
A witness described the woman with children, saying she was carrying two bags at the Diponegoro church. "Suddenly she hugged a civilian then (the bomb) exploded", the witness said.
Shattered glass and chunks of concrete littered the entrance of the Santa Maria Church, which was sealed off by armed police.
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The amendment was hastily reversed in 2017 after it was deemed by the previously little-known Islamist group to be blasphemous. Iqbal, who was shot in the arm Sunday evening during a rally in his constituency Narowal, is now under treatment in Lahore .
A street merchant outside the church said she was blown several meters by the blast. Meanwhile, two boys, whose initials are D and K, set out on their own riding a motorbike to Santa Maria's church. "One was wearing black trousers and one with a backpack", said the merchant, Samsia, who uses a single name.
"Police who arrived soon after the explosion shot the father to death because he had threatened to explode a bomb", Mangera told reporters.
In Jakarta, the Indonesian Church Association condemned the attacks.
"The main target is still security authorities, but we can say that there are alternative [targets] if the main targets are blocked", he told Reuters.
Indonesia, world's largest Muslim-majority country, has experienced a recent resurgence of local insurgent groups.
The bomb allegedly went off prematurely at 9 p.m., killing at least one of the tenants of the Wonocolo low-priced apartments behind the Taman Police station in Sidoarjo.
Forensic proof signifies the bombs have been connected to the attackers' our bodies, Karnavian stated.
Five members of Indonesia's elite anti-terrorism squad and a prisoner were killed in clashes that saw Islamist inmates take a guard hostage at a high-security jail on the outskirts of Jakarta. In recent years, the country has faced a new threat as the rise of IS in the Middle East invigorated local militant networks.
Indonesia's worst ever terror attack occurred in Bali in 2002 when bombs killed 202 people, mostly tourists.
Associated Press writers Niniek Karmini and Ali Kotarumalos in Jakarta, Indonesia and Stephen Wright in Bangkok contributed.