Happy to assist a fellow CCF member, I am, Jake. Here's what I can report, so far.
Brunk listed 27 different Parker counterstamps, and L.H. is not among them. The fact that it's as yet unlisted in his 2003 census is not unusual, and this doesn't typically enhance the value of a piece.
Over the past ten years, I've built an extensive online library of largely nineteenth century reference books; this, courtesy of Google. I found a half dozen listings for folks named L.H. Parker, ranging from the 1850's to 1890's. As anticipated, I'm unable to offer a slam-dunk attribution. That said, I do see one listing that is more likely than others, and further research might enhance this prospect.
The 1854 New England Business Directory and Gazetteer lists three businesses under the name of L.H. Parker. Two of them were in the creamery/butter business (Benton, NH & Woodsville, NH), and one was an electrician (Essex, CT). It's possible that the two NH listings were the same man, having businesses at multiple locations.
While the electrician is a more likely candidate in my eyes, given the nature of his work, I wouldn't rule out the other prospect(s). Given the letter style and the fact that counterstamping peaked during the decade of the 1850's, I'm inclined to put these Parkers at the top of the list. The electrician was more likely a member of the Masons, I tend to favor him. Of course, the dairy fellow(s) might also have been a Mason. Then too, either one might have had a stamp made if they were pursuing a patent. I'll look further down this avenue.
Sometimes, another specimen with a matching name and style will turn up; this, with an added town or occupation stamp. When this happens, former mavericks can be attributed. Such was the case with the Dr. G.G. Wilkins counterstamps that were ultimately attributed to Pittsfield, NH.