Asma and Nadeed Mohiuddin went to the supermarket in Beckton, east London, to buy nappies and stock up on supplies before their baby came.
But when they returned to their car they saw the penalty notice stuck on the windscreen.
Mr Mohiuddin said: "If you try and tell any heavily-pregnant woman they aren't accompanied by a child, you are going to get a very strong talking to.
"It's a big car park so if you got a space far away it would be a long way to walk."
Mr Mohiuddin said the free customer car park was 'half empty' and there were lots of other parent spaces free on the afternoon of February 23.
But parking operator Smart Parking rejected the couple's initial appeal, writing: "As the driver was not accompanied by a child, the vehicle was parked in breach of the advertised terms and conditions. We have no alternative but to uphold the Parking Charge Notice."
It was only when their story was reported in a local newspaper that supermarket bosses apologised.
The couple's baby girl, Aiza, had been due to be born on March 3 but Asma, 33, gave birth five days late on March 8. The couple have two other daughters, Haiqa, three, and Alayna, seven.Mr Mohiuddin said: "That we are not making it easier for heavily pregnant women to park is a national disgrace.
"On public transport designated seats are provided for the disabled, expectant mothers, and parents with children – there was never a distinction between the three.
"Someone needs to take the lead on this and set an example – and I'm sure it would be copied up and down the land. We can be doing a lot more for pregnant women."
An Asda spokesman said: "We would like to apologise to Mr and Mrs Mohiuddin for any inconvenience caused."
More on Parentdish: Get out of my space! The politics of parent and child spaces