This is what has happened in Wikipedia. Each page for a taxon lists the lineage to which it belongs (i.e., its parent, and its parent's parent, and so on), and also lists the children of that node. What this means is that if somebody edits the page for taxon A and adds taxon B as a child, they also need to edit the page for taxon B to make A its parent. If only one of these two edits is made the classification may end up internally inconsistent.
For example, the page for Amphibia lists the classification of Amphibia like this:
It also lists the child taxa of Amphibia:
So, the children of Amphibia are Temnospondyli, Lepospondyli, and Lissamphibia. Furthermore, Anura, Caudata, and Gymnophiona are children of Lissamphibia:
Given this, if I go to the pages for Anura, Caudata, and Gymnophiona I should see that each of these taxa lists Lissamphibia as its parent. However, only one of these (Caudata) does: the Anura and Gymnophiona both have Amphibia as their parents, not Lissamphibia.
The diagram below shows the taxa that have Amphibia as their parent:
Note that Stegocephalia have now turned up as an addition amphibian order, and that only Caudata is included in Lissamphibia. But what is striking is that another 274 Wikipedia taxon pages have Amphibia as their parent. These pages are all for fossil amphibians that do not fit easily in the existing Wikipedia classification.
From the perspective of building a database, the "has parent" relationship is the one I'd prefer to use, because that statement is going to be made just once (on the page for the taxon of interest). This seems a lot safer than making the statement "has child" on another page (for one thing, more than one page could claim a taxon as their child, which again will break the tree). But if we use the "has parent" relationship, our tree will be very bushy, with lots of fossil amphibian genera attached to the Amphibia node. This is going to make the tree hard to interpret, because this basal bush isn't saying that all these genera radiated off at once, but rather that we don't really know where in the amphibian tree these things go, so we'll have to settle for saying merely "they are amphibians" (for the cladistic theorists among you, this is Nelson and Platnick's "interpretation 2" in their "Multiple Branching in Cladograms: Two Interpretations", doi:10.2307/2412630).
So, the dilemma is whether to use "has child" relationships, and accept that these are likely to be inconsistent with the inverse "has parent" relationship, or use the "has parent" relationship, which will be internally consistent, but at the cost of potentially very large, unresolved bushes due to fossil taxa of uncertain affinities.