The answer is YES! Absolutely!
Because of the random way in which autosomal DNA is inherited, each child only receives a portion of each parent’s DNA. The child is made up of 50% from the mother and 50% from the father but if no parents are tested, matches for that other 50% of each parents’ DNA will not be represented. That is why it is so important to test parents first, especially if funds are limited and age or health could be a factor.
At Family Tree DNA, my mother has 290 Family Finder matches in the database as of today. Applying the “in common with” filter and selecting my name, there are only 116 matches. Of my mother’s 290 matches, my results only reflect 116 of them. This means that 174 maternal matches would have never been discovered without testing my mother. Applying the “in common with” between my mother and sister only yields 108 matches, which means my sister would have lost even more genetic data than I would have if my mother and I had not tested. Keep in mind, those “in common with” match lists that my sister and I have are not identical. We have each inherited different bits of autosomal DNA and will, therefore, have some matches in common and some different.
The reason that I tested my sister is because my father is deceased. If I had a complete set of parental autosomal DNA, I might do my own autosomal DNA test for entertainment, but there would not be a significant value to help me reconstruct my genetic tree. Since my father is not living, I cannot get the full picture, as I can with my mother. I have to use the “in common with” matches that my sister and I have. I have to do this backwards, by subtracting out what belongs to our mother.
In my own results on the Matching page, I apply the “not in common” filter and select my mother. Any of the matches who match me but do not match my mother must come from my father by default. I can then repeat that process using my sister’s results. Merging our two results together can provide us with a list of matches that came from our father. This is not a perfect solution, but it’s a step in the right direction. If we had more siblings, it would be even better.
The bottom line: My recommendation is that if both parents are living, both parents should be tested. If at least one parent is deceased or otherwise unavailable to test, then the living parent and as many living children as possible should be tested to get a more complete snapshot.