If you look at the line point attribute table, you'll notice there is a 'ParcelID' field. That field contains the ObjectID of the Parcel that the line point is related to. However, users have no control over how this system managed field is populated, nor can they edit this field if desired. When creating a line point manually, all the user has the ability to enter is the from point and the to point of the line the line point should snap to - the ParcelID field is populated for your automatically. I believe the ParcelID field is populated with the top most Parcel in the event of overlapping Parcels. I think the same rule is applied during a Join process (the top Parcel is used to populate the ParcelID field for any line points created during the join).
There are a few reasons I would like to see the ability to control what Parcel a line point is related to. The first and most important is that accuracy for line points is inherited from the Parcel Layer (not the line layer). That means that if the Parcel the line point is related to has an accuracy of 5, the line points will have an accuracy of 5. However, since I have no control over what Parcel the line point gets related to, I essentially have no control over the accuracy value for the line points.
The second reason is a rarer scenario, but if I delete a Parcel that several line points are related to (via the ParcelID field), the line points will be deleted as well. This occurs even if there are other parcels in the area that require line points to be present. So, even though the 'FromPointID' and 'ToPointID' values in the line point attribute table still exist, because the ParcelID has been deleted, the line points will be deleted.
Having the ability to set what Parcel line points are related to will give users more control over line point behaviour in their Parcel Fabrics.