What we can be certain of is the Daily Mail would have paid the family for their story and they would've been paraphrased for by the journalist for "narrative impact"
You can be far from certain of that.
It appears the story was picked up first by the Blackpool Gazette - the parents went to their local paper as local people often do.
The next stage is it was picked up and hawked around by a news agency - not a shop, a business involved in selling stories to other media outlets.
The evidence for that is the copyright imprint on the pic - Ross Parry, an established agency.
It is possible the parents received some money, but if they did, it is unlikely they were paid directly by the Mail.
Further evidence of it being a free for all is no one has an exclusive, all the media outlets have the same stuff.
If the Mail - or anyone else - paid - they would want exclusivity, or at least an exclusive line.
As regards paraphrasing for impact, the words are direct quotes and the mother speaks on the video, again in her own words.
Other than the above, your post is entirely accurate and therefore could appear on Mail Online.