*In South Australia you can be prosecuted for leaving your kids in the car at a petrol station, or if you go into a shop.
You can also be prosecuted for letting them to school alone, too young, as you can if you leave them home too young.*
Out of all of Australia's states and territories SA is one of the ones where this is the least likely to be possible. There simply is no legislation covering anything like this at all.
There is criminal legislation that is relevant only if serious harm or death results, but that's absolutely not even near the same ball park as no serious harm or death occurring while children stay in the car at the servo or while you go into a shop or a while a child walks to school alone:
Criminal Law Consolidation Act 1935
^^14—Criminal liability for neglect where death or serious harm results from unlawful act
(1) A person (the defendant) is guilty of the offence of criminal neglect if—
(a) a child or a vulnerable adult (the victim) dies or suffers serious harm as a
result of an unlawful act; and
(b) the defendant had, at the time of the act, a duty of care to the victim; and
(c) the defendant was, or ought to have been, aware that there was an appreciable risk that serious harm would be caused to the victim by the unlawful act; and
(d) the defendant failed to take steps that he or she could reasonably be expected to have taken in the circumstances to protect the victim from harm and the defendant's failure to do so was, in the circumstances, so serious that a criminal penalty is warranted.
Maximum penalty:
(a) where the victim dies—imprisonment for 15 years; or
(b) where the victim suffers serious harm—imprisonment for 5 years.
(2) If a jury considering a charge of criminal neglect against a defendant finds that—
(a) there is reasonable doubt as to the identity of the person who committed the unlawful act that caused the victim's death or serious harm; but
(b) the unlawful act can only have been the act of the defendant or some other person who, on the evidence, may have committed the unlawful act,
the jury may find the defendant guilty of the charge of criminal neglect even though of the opinion that the unlawful act may have been the act of the defendant.
(3) For the purposes of this section, the defendant has a duty of care to the victim if the defendant is a parent or guardian of the victim or has assumed responsibility for the victim's care.
(4) In this section— act includes—-
(a) an omission; and
(b) a course of conduct;
child means a person under 16 years of age;
cognitive impairment includes—
(a) a developmental disability (including, for example, an intellectual disability,
Down syndrome, cerebral palsy or an autistic spectrum disorder);
(b) an acquired disability as a result of illness or injury (including, for example, dementia, a traumatic brain injury or a neurological disorder);
(c) a mental illness;
serious harm means—
(a) harm that endangers, or is likely to endanger, a person's life; or
(b) harm that consists of, or is likely to result in, loss of, or serious and protracted impairment of, a part of the body or a physical or mental function; or
(c) harm that consists of, or is likely to result in, serious disfigurement;
unlawful—an act is unlawful if it—
(a) constitutes an offence; or
(b) would constitute an offence if committed by an adult of full legal capacity;
vulnerable adult means a person aged 16 years or above whose ability to protect himself or herself from an unlawful act is significantly impaired through physical disability, cognitive impairment, illness or infiermity.
www.legislation.sa.gov.au/LZ/C/A/CRIMINAL%20LAW%20CONSOLIDATION%20ACT%201935/CURRENT/1935.2252.UN.PDF#page29
The only other peripherally relevant legislation in SA deals with removing children from dangerous situations but it has nothing at all to do with anyone being prosecuted for anything:
^^Children's Protection Act 1993
16—Power to remove children from dangerous situations
(1) If an officer believes on reasonable grounds that a child is in a situation of serious danger and that it is necessary to remove the child from that situation in order to protect the child from harm (or further harm), the officer may remove the child from any premises or place, using such force (including breaking into premises) as is reasonably necessary for the purpose.
(1a) If—
(a) a restraining order has been made under section 99AAC of the Summary
Procedure Act 1921 preventing a person from residing with a child; and
(b) the child is residing, during the operation of the order, with the person,
the child will be taken, for the purposes of this section, to be in a situation of serious danger from which an officer is authorised to remove him or her under subsection (1).
(2) An officer's powers under this section are subject to the following limitations:
(a) a police officer below the rank of inspector may only remove a child from a situation of danger with the prior approval of a police officer of or above the rank of inspector unless he or she believes on reasonable grounds that the delay involved in seeking such an approval would prejudice the child's safety;
(b) an employee of the Department may only remove a child from the custody of a guardian with the Chief Executive's prior approval.
(3) An officer who removes a child under this section must, if possible, return the child to the child's home unless—
(a) the child is a child who is under the guardianship, or in the custody, of the Minister; or
(b) the officer is of the opinion that it would not be in the best interests of the child to return home.
(4) If an officer removes a child under this section, and the child is not returned to the child's home under subsection (3), the officer must deliver the child into the care of such person as the Chief Executive, or the Chief Executive's nominee, directs.
(5) If the Minister does not already have custody of a child who is removed from a situation of danger under this section, the Minister has custody of the child until—
(a) the end of the working day following the day on which the child was removed; or
(b) the child's return home,
(whichever is the earlier).
(6) The power to remove a child under this section is in addition to, and does not derogate from, the powers of an authorised police officer under section 51(4).
www.legislation.sa.gov.au/LZ/C/A/CHILDRENS%20PROTECTION%20ACT%201993/CURRENT/1993.93.UN.PDF#page22