A search on EEBO for A parte of a register leads to STC (2nd ed.) / 10400, dated by EEBO 1593. There are two image sets, STC / 364:03 and STC / 506:09.

STC / 364:03 bears the title A demonstration of the truth of that discipline, which Christ hath prescribed in his Word, for the government of his church, in all times and places, until the end of the world, and is undated. Claire Cross in her ODNB entry on John Udall mentions this work, implying that Udall wrote it. A search on the title A demonstration of the truth of that discipline leads to a different result, STC (2nd ed.) / 24499, dated by EEBO 1588. A note on the first leaf gives the date 1589 in a 16th or 17th century hand. It is not the same edition as 364:03. Udall’s name is written on the title page of 24499 in a 16th or 17th century hand.

STC / 506:09 actually has the title A parte of a register. Cross does not mention this work (a compilation of 41 different documents according to its own Table of Contents, none of which has the title A demonstration of the truth of that discipline) and it is not clear what connection A parte of a register might have with Udall.

Peter White: The image set at reel position STC 506:09 shows A demonstration of the truth in the  context of A parte of a register; Udall's A demonstration begins on Document Image number 284 of 327 of the set of images at STC / 506:09.

Anna Battigelli:  It is interesting that though Udall's A Demonstration appears last in the volume, it is not listed in the table of contents for A parte of a register.  In its place in the TOC is "A Prayer for the Faithful," which appears nowhere in the volume.  Could this be a conscious act of misdirection, both concealing and calling attention to Udall's piece?