There's a bit of a (completely understandable) misconception at the heart of this question - on the most part it wasn't parents sending their children to the countryside, but rather a massive state-directed effort to remove the children from the way of the bombs. That aerial bombardment would be a central part of the war had been known about for some time before war broke out; As far back as 1932 Prime Minister Stanley Baldwin stated that “The bomber will always get through” – the fear that anti-aircraft technology was simply not capable of preventing mass bombing attacks. This fear of bombing was increased by the Bombing of Guernica in 1936, with the Luftwaffe reducing the Spanish republican stronghold to rubble, with many civilian casualties.

Accordingly, on the weekend of the 2nd of September, the entire rail network was utilised to shepherd hundreds of thousands of children away from London, and other major urban centres expected to be bombing targets. This article by the Imperial War Museum has a poster from Southern Railways advertising how normal rail services were suspended in order to facilitate the Evacuation. The article also shows a picture of the pupils of Myrtle school gathered for the evacuation – because it was mostly done via schools, with entire schools taken to train stations and packed off to rural locations that were assumed to be safer.

A popular media representation of this process is The Lion, the Witch and the Wardrobe, with the Pevensies evacuated from London to the countryside – without their parents. The book’s author, C.S Lewis, had taken in evacuees from London into his house during the war, and so would have been familiar with the process. (Side note: The sight of children at train stations, with their few possessions and a name tag, would also be the inspiration for Paddington Bear)

At it’s most extreme, some children were sent oversees via the ‘Children's Overseas Reception Board’ which was a government-sponsored programme to remove children to country’s of the British Empire. The scheme was halted when a ship headed for Canada (SS City of Benares), was sunk by a German submarine, with the tragic loss of most of the evacuee children on board. When in the countryside, children were taken in by strangers, who received a state allowance for their upkeep – because it was a government scheme to remove the children, and not the general population.

This contemporary article describes how London suddenly became virtually childless, in a weekend, and how parents and children reacted. It’s also worth noting that the article cites many anxieties about future bombing, but also sometimes a blasé attitude, and of people who don’t think it will be ‘that bad.’ It’s important to note that while the children were evacuated on the 2nd-3rd September 1939, the first bombing of London (and the UK) was the 7th September 1940 – a full year later. Other cities that were heavily bombed, such as Liverpool and Coventry were bombed even later – with the latter heavily bombed in November 1940. Accordingly, parents with jobs and homes in the city would have lived mostly as normal during the so-called ‘phony war’ – and not felt the need to leave the cities. During this period, many evacuee children also returned home. When the blitz started in earnest, many would be subsequently ‘re-evacuated.’

Thus, as the answer to your question, the parents didn’t go with their children, because it was largely a government scheme. The government directed the rail network, to take the children out of the cities or even in some cases abroad.