I assume you have already been introduced to Mountain Prominence - Definition, and know that every mountain has exactly one "key saddle" joining it to higher terrain. The line parent is just the first peak you come to on that "higher terrain". A loose definition of a "line parent" is that it is the first higher peak you come to after the key saddle. In other words, you go down to the "key saddle" of the peak, and then follow the ridge upwards till you come to a higher peak.
But the above definition obviously depends on what bumps a given database designates as "peaks". You might start out with one peak being the line parent, but if you add an intermediate peak later, then you might want to update the line parent. For example, suppose you've got two peaks, Peak A is 3000m and its parent is Peak C which is 3200m. But later on, you also decide to insert Peak B which is 3100m into the database, and lies on the same ridge between A and C. Is your original line parent for A now "wrong", just because your database now includes the bump "B"? We say no. Your database is not wrong because it still designates C as the parent of A. You could revise it, but the original is not wrong, because there could be other databases that don't contain B.
The only way a bunch of databases could completely agree on a certain line parent for every peak is if they all agreed on a prominence threshold. Eg: You only consider peaks with 300m prominence as potential line parents. But having a single prominence threshold is not appropriate for all of North America, so this is not typically done in any database. What to do? This is the discussion of this document.
CONCLUSION: As of March 2012, it is now clear that when trying to reconcile two databases, there is no point in trying to make the line parents agree, unless both databases also define a prominence threshold. "Line Parent" is just a convenient method to prove the prominence. It is generally the nearest higher peak at the time the prominence was determined. For peaks with distant parents, it is usually a significant named peak, not a lesser unnamed peak.
The only thing that is definitive about a peak is its height and its prominence, and its key saddle. But not its line parent.
For example, suppose that at the time the prominence was figured out, a given database had Peak C as the parent of A. If at a later time, an intermediate peak B is inserted into the database, and it is also higher than A. It is still acceptable to have that C is the line parent of A. It is not regarded as a database mistake.
When reconciling two databases, there is no point in trying to make the line parent info agree.
Another case where the "line parent" on file can vary is when a peak has a distant parent, we typically "skip over" a less significant peak as you approach a better known peak. For example, in the Canadian Rockies, the parent of Mount Joffre can be said to be Assiniboine. Assiniboine is about 50 km from Joffre. If you start traversing the ridge from Joffre toward Assiniboine, you don't come to higher ground until you start going up the final south ridge of Assiniboine. But less than 1 km before the summit of Assiniboine, you encounter a bump called Lunette Peak, which itself has only 28 m of prominence. Should Lunette be the "line parent" of Assiniboine? No, because it is a low prominence subpeak of Assiniboine.
The reason Lunette is skipped is because it is "almost" to Assniboine. If had been only half way to Assiniboine then it would be the proper "line parent" of Joffre.