A few blogs ago, vlorbik sent a comment that included the suggestion that "mother function" might have been one of the victims of social censure and thus gave rise to the emergence of the (apparently) more socially acceptable "parent function."
From what I have been able to discover in a short period of researchs, the use of terms like "parent function" seems to have worked its way into mathematics from statistics, which seems to have gotten it from the anthropologists/sociologists.
Across America and Asia: Notes of a Five Years' Journey Around the World ... - Page 250 by Raphael Pumpelly - Voyages around the world - 1870
"if it should bear the same relation to the parent population that.."
Prior to 1900 there are almost no listings of "parent function" in a mathematical usage. Around the end of the 19th century, statisticians began to talk about the distribution from which a sample was taken as the "parent distribution" of the sampling distribution (population of all samples of some size n). Some of these may be related to the study of eugenics in which the study was about the relations of some characteristic of the offspring to the actual parents, but the usage grew.
Data reduction and error analysis for the physical sciences , by Philip R. Bevington - 1969 10-2 the F TEST As discussed in the previous section, the x2(chi square) test is somewhat ambiguous unless the form of the parent function is known because the statistic
x2 (chi-sq)... "
Occasionally I find the term "parent function" applied in this way when the distribution of the original sample was a normal distribution.
The Annals of mathematical statistics - Page 179
by American Statistical Association, Institute of Mathematical Statistics, JSTOR (Organization) - 1948" Usually the parent function is the Type A or normal curve, as discussed by Gram "
There are also some early uses of the "parent function" in association with the use of inverses and derivatives in calculus and analysis texts back to about 1925. By 1970 the term had become commonly understood, but not abundantly used.
The teaching of secondary mathematics - Page 521
by Charles Henry Butler, Frank Lynwood Wren - Education - 1965 - 613 pages
".. of an inverse function and its relation to the parent function or else in failure to attach clear meanings to the terminology and notation employed. ..."
It was the 1980's and the introduction of computers and graphing calculators into modern classrooms that seemed to make the term "parent function" ubiquitous. Any function that appeared on the calculator was a parent function, and the translations, rotations, shears, etc became the "children".
I think this use also led to the introduction of "mother function" rather than the other way around. I can only find a few examples of "mother function" and there does not seem to be any pattern to the frequency as one might expect if a term had arisen to replace this one as an "off-color" predecessor. In fact, it seems "mother function" is more commonly used by continental writers, often in conjuction with "daughter functions"; but admittedly the sample size I have to draw on was small.
If you are one of those people with access to old journals, or a collection of old texts, I would appreciate any references to the use of any of these terms and a source earlier than 1900; and if you have a way to make a digital copy and send it by email, I will have my students name their children after you.