Maggie Black, who was five years old, died suddenly on December 1 after becoming unwell.
Her family told of how an ambulance dispatched to treat the St John's Primary School pupil took an hour and ten minutes to arrive.
The Blacks have started a petition for the Northern Ireland Fire and Rescue Service to to be dispatched to emergency calls in Carnlough, Glenarm, the Glens of Antrim and all rural areas in Northern Ireland.
As of Tuesday morning, the petition (available here) had gathered over 13,000 signatures in just two days.
Maggie's family wrote: "On the morning of 1st December 2021, our lives changed forever, as we lost our innocent, fun-loving, beautiful “Wee Maggie”. She was the light of our lives, who made an impression on everyone she met. We as a family will never be able to put into words the impact of that morning’s events.
"Whilst we are under no illusion about the pressures and strains on the ambulance service, it is surely unacceptable that we had to wait one hour and ten minutes for an ambulance to arrive when our five-year-old daughter’s life was at stake. Fortunately, we contacted a family member, a local firefighter, to come to the scene; he then dispatched the fire crew.
"We are truly grateful for the medical expertise, professionalism and kindness demonstrated by the fire service. As first responders with lifesaving equipment we know that Maggie was given the best available support, this is the greatest comfort to us at this time."
'Difference between life and death'
"We as a family and our community are truly shocked, worried and saddened that the dispatch of the fire service is not an automatic response in these situations. Sadly, not everyone in our community will get access to this emergency medical assistance during the crucial 'golden hour'. This can ultimately mean the difference between life and death."
The family went on to call for support from the wider community to guarantee "the mandatory dispatch of the fire service in emergency situations to support our vulnerable rural families and ultimately give us all the best chance of life."
Commenting on the petition, a spokesperson for the Department of Health (DoH) said: "The DoH, the Northern Ireland Ambulance Service (NIAS), and the Northern Ireland Fire and Rescue Service (NIFRS) extend their sincere condolences to the Black family and fully support any initiative to improve emergency medical cover in rural communities.
"As part of the emergency medical response to rural communities, a number of community responder schemes are in place across Northern Ireland which can be dispatched by NIAS alongside paramedics as required; however, there are currently none covering the Glenarm area.
"Whilst these community volunteers generally do not attend incidents involving children under the age of 12, due to the specialist training and equipment required, NIAS would welcome approaches from any group of volunteers who express an interest in creating new local groups, including in the Glens of Antrim region."
The NIAS and NIFRS are currently working together to progress the development of a regional joint response model, the spokesperson added: "With regard to potential collaboration with NIFRS, NIAS previously ran a promising co-responder pilot in which NIFRS crews were dispatched to suspected cardiac emergency calls in the local catchment area of fire stations in addition to the normal emergency ambulance response.
"NIAS and NIFRS met in November 2021 to consider how this model might be rolled out across Northern Ireland, and both organisations are working closely together to progress the development of a regional joint response model."
Councillors encouraged to sign petition
At Monday night's meeting of Mid and East Antrim Borough Council, elected members spoke in support of the Black family's efforts.
Coast Road Sinn Fein Councillor, James McKeown said: "I would encourage as many councillors as possible to sign the petition."
His comment was echoed by Mayor, Cllr William McCaughey, who added: "The sooner we can get trained emergency personnel to anywhere in our borough is more than welcome."
Coast Road Alliance councillor Alderman Gerardine Mulvenna described the death of Maggie as "very tragic and very sad".
The Mayor reiterated his sympathy to the Black family on the "sad passing" of their daughter Maggie.