Trailers for horror films were played before a screening of the children's film Peppa Pig: Festival of Fun.
Families at the Empire Cinema in Ipswich said their children were left in tears after seeing adverts for psychological horror Ma and scary superhero flick Brightburn.
Parents removed their children from the auditorium until the trailers stopped.
Empire Cinemas has apologised for "any distress caused" and said it is investigating.
The trailer for Ma features dead bodies, sexual violence and a man being hit by a car, while Brightburn features a child with a scary mask as well as blood and violence.
'Confused and crying'
Charlie Jones, from Manningtree in Essex, took daughter Annie, two, for her first cinema trip, along with 10-month-old Fred.
Mrs Jones, a BBC journalist, said Annie had been subdued since the experience and had asked about a mask seen in one of the trailers.
She said: "Normally I would expect her to be singing and dancing when watching something like [Peppa Pig] but she was just really subdued.
"I tried to cover her eyes during the trailers and told her they were silly films for mummies and daddies, but there were lots of kids crying and she was very confused and started crying too."
Mrs Jones, 34, said: "I went mad at [the manager]. It was meant to be a special day for Annie. I don't think he understands the impact of it.
"Annie is very imaginative and scared of monsters and things like that at the moment.
"You go to the cinema and you expect it to be a safe family day out, you don't expect her to be exposed to anything which could do harm."
A spokeswoman for Empire Cinemas said: "As soon as the staff on site were made aware of the situation, the programme was stopped and trailers were taken off-screen immediately.
"We do sincerely apologise for this and for any distress caused and will be reviewing our internal procedures to ascertain how this came to be."