Campbell's father, Dirk, told the BBC she "wanted to create a better world and she would do everything in her power to do that".
He added: "I told her of course that she was putting her life in danger, which she knew full well she was doing. I feel I should have done more to persuade her to come back, but she was completely adamant."
The YPJ is an all-female brigade of the Kurdish People's Protection Units YPG, which has around 50,000 Kurdish men and women fighting against Isis in northern Syria.
The group has been defending the Kurdish-majority city of Afrin from Turkish forces backed by Syrian rebels after they launched an offensive in mid-January.
Ankara considers the YPG a terrorist group, an offshoot of the outlawed Kurdistan Workers' Party (PKK) which has led an insurgency in Turkey for decades.
Campbell had been qualified plumber living in Bristol before she left to join the Kurdish militia in Syria.
Her friends and commanders are thought to have tried to dissuade her from going, saying it was too dangerous.
Her father said his daughter was an "incredibly principled, brave, determined, committed woman" whose death had left him "in pieces".
"She was determined to live in a way that made a difference to the world and she was determined to act on that and do whatever it took," he told the BBC.
"She was prepared to put her life on the line. There aren't many people who do that.
"In retrospect I think that I probably should have done more to dissuade her (from going to Syria) but I also knew that she would never have forgiven me if I had actively prevented her from going.
"I couldn't affect or try to influence her own perceived destiny. It was the most important thing in life for her."
Macer Gifford, a British volunteer with the YPG, said: "I travelled into Rojava with Anna in May 2017. She was kind, funny and brimming with energy. She had heard about the bravery of the YPJ and they inspired her to come out to fight for democracy, equality and peace in Syria.
"She wanted to fight Isis and to rid Syria of the evil that has ripped the country apart," said Gifford, who uses a pseudonym. "Anna then bravely volunteered to join the defence of Afrin. She bravely fought alongside her YPJ sisters, they opened corridors for civilians to leave and delayed the invaders long enough to save thousands of lives."
Mark Campbell, co-chairman of the Kurdistan Solidarity Campaign, said: "Anna is a woman who seemed to have more humanity in her little finger than the whole of the international community.
"She is an inspiration and a hero."
He said Campbell, who is no relation to him, was killed alongside two Kurdish women amid the air strikes.
He added: "I did not know her but I met with her father this morning. I have the utmost respect and condolences for her family."
In a statement, YPJ commander and spokeswoman Nesrin Abdullah said Campbell's death was a "great loss".
She said: "Campbell's martyrdom is a great loss to us because with her international soul, her revolutionary spirit, which demonstrated the power of women, she expressed her will in all her actions.
"On behalf of the Women's Defence Units YPJ, we express our deepest condolences to (her) family and we promise to follow the path she took up. We will represent her in the entirety of our struggles."
Conflict between Turkey and Kurdish groups has been inflamed since January.
Over the weekend, Turkey's president Recep Tayyip Erdogan said the country's military had captured the town centre of Afrin, which was previously controlled by the YPG.
