The family was celebrating their son's sixth birthday and had spent hundreds of pounds on tickets, a friend said.
Mr Singh, from Coventry was in breach of health and safety rules when he refused to exchange his Kirpan for a replacement necklace, the park said.
Sikh elders advised that could be worn as an acceptable substitute, it added.
Kirpans have not been allowed to be worn in the park for "a number of years" because of health and safety risks, the park said, adding, it was "regrettable" the family did not take up their offer and were denied entry.
The dagger, or sword, is one of five Ks initiated, or baptised, Sikhs wear.
The others are Kesh (uncut hair), Kara (a steel bracelet), Kanga (a wooden comb) and Kaccha - also spelt, Kachh, Kachera - (cotton underwear).
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The family friend described what happened at the park near Tamworth, Staffordshire on 2 June in a blog under the name of The Secret Psychiatrist.
The friend said they were "ashamed" and "disgusted" by the incident that had spoiled a happy family day out.
The Sikh Press Association, which is liaising with the family and a Sikh legal advocacy team, said together they were calling for the park's rules to be changed.
"In this day and age where the Kirpan can be taken into Parliament, it is disappointing to see an amusement park ban this article of faith from their grounds," a spokesman said.
Although Sikhs can choose to substitute their Kirpans, the park's offer devalued what the five Ks meant, he added.
"Sikhs treat them like their own limb.
"The word (Kirpan) means 'a blessing' so it's not just a physical item, and to use something that has probably been lying in a cupboard, it just devalues it."
In a statement the park said as well as consulting Sikhs, an independent health and safety report confirmed that wearing a sheathed dagger, or anything similar, "posed a viable compromise to safety" on a ride.
"It is for this reason that cameras and other similar articles are also forbidden on rides. The same rule applies to all visitors without exception."
A Sikh family was denied entry to Drayton Manor Theme Park after an adult refused to remove his ceremonial dagger, worn as a symbol of his faith.