When Apartheid became formalised there was not really a scientific way of determining ones racial classification, as we can do with genetic testing today. The classifications that were decided under the Population Registration Act 1950 were white, coloured, and native with later classification granted for Indians and other Asians. Whites, coloureds and native bantus already existed as cultural and racial identities with most people identifying as such so there was not a huge issue in determining ones racial classification. However that is not to say there were no issues. The biggest point of contention was where the line was drawn between white and coloured, and coloured and black.

First a bit of background on the definition of 'coloured' as the word has a unique meaning in South Africa, and is not considered as derogatory or old fashioned as in the United States or United Kingdom. Coloureds are the descendants of mixed relations between European and the Khoi San, Bantus, and Asians, primarily Malays. Through time they became a separate ethnic group and identity. The language of Afrikaans, used by the majority of the white and coloured population in South Africa today actually grew out of the pidgin Dutch that the coloureds would speak. Malay words such as ''piesang'' for banana was adopted by Afrikaans, and the infamous and condemned term of ''kaffir'', used to disparage the black population, was descended from the Arabic word ''kuffar'' which was widely used by the the Muslim Malays. Cape Malay still remains as a somewhat distinct identity within the coloured community, however the vast majority of coloureds identify simply as coloured. They are distinct from the white and black population, and see themselves as such, with a rich culture and heritage and a tragic history. The best way to describe it would be as a mixed race ethnic group, somewhat comparable to the use of term 'mulatto' in the Americas.

Now the line between white, coloured, and black was often blurred. Genetics are a bit of a lottery so it was very possible that a white person could have darker skin than a coloured, and a coloured could have darker skin than a native Bantu. For the most part this wasn't an issue, as other social indicators and language could be used to come to a classification. Where you lived, your occupation, you wider social circle and family as well as your language were common indicators. A person might be light skinned, but if they were born to an Xhosa mother and father, spoke Xhosa and lived in Transkei then they were for all intents and purposes a native Bantu regardless of their skin colour. That said there were noticeable areas were identity overlapped, especially between those of white Afrikaans and coloured identity. Where the lines were blurred on someone's race, the authorities used a series of pseudo-scientific methods to determine someone's racial classification, the most notorious being the so called ''pencil-test''. The concept around this came from the fact that most coloureds had inherited the gene for curly hair from their Bantu heritage, and as a result if you stuck a pencil in their hair it would stick. This was a degrading and ridiculously stupid way of determining someone's race, but was used by South African authorities in cases of contention. Where the issue was related to whether someone was black or coloured they would be asked to stick a pencil in their and shake their head, if it fell out they were coloured and if it stay in they were black. It was absolutely absurd. Amongst some lighter skinned coloureds, and lighter skinned bantus, there was a benefit to passing as white or coloured, so many people would try and reclassify their race to improve their opportunity and rights however this was a difficult process that did not solely rely on the ''pencil-test''.

Now on to someone more noticeable cases of contention. Sandra Laing was born to two white Afrikaans parents. She was always dark skinned but as she grew older this became more noticeable, with darker skin and curlier hair, with her being shunned by the community. This was a result of a dormant gene for darker skin that came from distant African heritage, rather than an illegitimate relationship (and yes, having sex across racial lines was illegal but that is another issues entirely). In fact Afrikaners actually have a not insignificant amount of black and Asian ancestry, making up to 5% of their genome, further bringing into question who exactly is white? Anyway, Sandra was expelled from her white school, and her parents underwent a massive legal battle to have her reclassified as white and were finally successful. However she spent time in a coloured boarding school in between, and subsequently went to live a life with the black population despite still be classified as white. Her children were taken into the care of the state due to her refusal to re-identify as ''coloured'', the same race as her children. She lived without reconciling with her conservative father who renounced her for running away with a black man, but she did reconcile with her mother but still identifies as black feeling more accepted and comfortable with that identity.

Dimitri Tasfendas was born in Mozambique to a Greek father and mixed Mozambican mother. He struggled with his mixed identity, often discriminated for it as a child, and went on to become a merchant seaman. His life was rather interesting, with him being involved with various communist movements as he found himself around the world. He ended up in South Africa, where he had spent time as a child, and was classified as white due to his Greek heritage. Despite this he fell in love with a coloured girl, but relations between races were forbidden. He applied to change his identity from white to coloured, but it was rejected. This would be the typical depressing end of a tragic love story, destroyed by the narrow mindedness of Apartheid, if it was not for the fact of where he worked. His job was as a messenger in the South African parliament. His motives are disputed, with him eventually be deemed insane and blaming it on psychosis related to a worm in his stomach, but he was responsible for the assassination of Prime Minister Hendrik Verweord, often called the architect of Apartheid.

And although this sub does not particularly like personal anecdotes I hope this one will be allowed. My parents have always told me the story of a family friend who was head over heels for a girl during the late 1980s. She had beautiful dark eyes, and a slight tan which really wasn't unusual for South Africa. She spoke perfect Afrikaans, so he presumed she was an Afrikaner rather than Portuguese which were a noticeable minority in South Africa. For all intents and purposes she was white. He had met her at a club, and was madly infatuated with her. As things progressed he went to visit her parents. To his surprise he saw dark skinned parents, who were obviously coloured, and a completely different skin tone to their daughter. Although mixed marriages were now legal, with the law against them being repelled in 1985, there was no prospect that this man's conservative family would accept such a marriage. Things quickly ended between them, a tragic and unnecessary consequence of the legacy of apartheid in South Africa and the taboo and narrow minded views of race.

Very interesting question. Racial identity is very fluid, and where the lines became blurred the Apartheid authorities struggled to deal with it, resulting in ridiculous measures such as the ''pencil-test''. People, who through the lottery of life were born dark skinned to white parents were at times reclassified as coloured. But it must be said for the most part this wasn't a huge issue as the majority of the country conformed to existing racial classifications. To directly answer your question, the children would usually be classified as the same of their parents, but not without issue.

Sources and further reading:

R.W. Johnson - South Africa: The First Man, The Last Nation

Rory Caroll - The Black Woman, With White Parents - The Guardian https://www.theguardian.com/theguardian/2003/mar/17/features11.g2

Alan Cowell - South Africa's Coloreds: A Group Torn Between Black And White Worlds - New York Times https://www.nytimes.com/1985/09/11/world/south-africa-s-coloreds-a-group-torn-between-black-and-white-worlds.html

Population Registration Act 1950 - South African Parliament https://www.sahistory.org.za/sites/default/files/DC/leg19500707.028.020.030/leg19500707.028.020.030.pdf

Deborah Posel - What's In a Name? Racial Categorisations Under Apartheid And Their Afterlife - Transformation Journal (47) 2001 accesible via http://pdfproc.lib.msu.edu/?file=/DMC/African%20Journals/pdfs/transformation/tran047/tran047005.pdf

David Beresford - Obituary: Dimitry Tsafendas - The Guardian https://www.theguardian.com/news/1999/oct/11/guardianobituaries.davidberesford

Hollfelder et al., Patterns of African and Asian admixture in the Afrikaner population of South Africa https://www.biorxiv.org/content/10.1101/542761v1.full.pdf