It’s a rare designation that is currently only issued in 3 states: Iowa, Delaware and Minnesota. Some other states previously had an LPA designation but have since stopped issuing it.
(UPDATE 11/3/15: See the comment below from Jason Jones. The LPA is a rare designation, but the designation or some equivalent of it, is available in several other states besides Delaware, Iowa and Minnesota. See his comment for more info.)
In Iowa, one becomes an LPA by passing the 2-part Accredited Business Accountant exam. Iowa allows LPAs to practice as a public accountant and LPAs can do all of the things a CPA can do, except for audits of financial statements.
For tax purposes, the Iowa LPA designation is not recognized by the IRS (even though Part 2 of the ABA exam is devoted 100% to taxes).
LPAs are a rare breed. At the time I got my LPA designation in 2013, I was told informally that I was the 126th active LPA in the state.
For more tax and accounting terms, visit the Glossary page.