What does BPE/PE in some archaeology journals stand for?

BPE means Before Present Era. Now we know the words behind the acronym, we still don't know the meaning of those words.

21.12 dates

Note:
non-religious alternatives

to BC (before Christ) and AD (anno domini – year of our Lord)
are BCE (before common era) and CE (common era),
and also BPE (before present era) and PE (present era).
However these acronyms are not, as yet, very common.
[…]

–– Adrian Wallwork: "English for Academic Research: Grammar, Usage and Style", Springer: New York, Heidelberg, 2015.

As used in the question's quote apparently. Here is another usage of the acronym, but with one possible definition that's different:

Before the Present Era (BPE): a dating system based on the use of radiocarbon-dating, which uses January 1, 1950, as its baseline. Therefore, 10,000 years BPE equals 10,000 years before New Year’s Day, 1950.

–– Canadian History: Pre-Confederation

That is

BC = BCE ≠ BPE = BP

We see that originally bp was the archaeological acronym to refer to radiocarbon dating – which is no longer useful for objects created after we engaged in nuclear follies. One addition might therefore be useful though: the E in BPE seems to appear mostly because '1950' is no longer really 'present history'? But that is quite unfortunate, as 'era' in a calendar sense is also well established in one reading as the period commonly abbreviated AD.

So we see in the quoted article that BPE means Before Present Era as well. This is to avoid the association with Christianity. But here it means exactly the same thing. Different to BP Before Present, as that 'Present' is everything after when we started littering the atmosphere with radionuclide fallout.

As it did become clear to OP, from the context, like

Gold in early Southeast Asia
Gold is fairly widely, though irregularly, distributed throughout Southeast Asia in igneous and metamorphic hard rock deposits and in sedimentary placer deposits. The region was known to the Indian merchants of the 1st millennium BPE… Gold first appears in the archaeological record in 400 BPE, at about the same time as iron, semiprecious stone polishing and glass working, suggesting that the techniques of gold extraction and working were quite plausibly introduced to Southeast Asia via Indian and/or Chinese merchants seeking gold ores.

it then becomes clear to all that for that article the usage seems to be:

BC = BCE = BPE ≠ BP

As there is a difference of 1950 years to observe: That is again unfortunate as this BPE is not a very well established acronym and it lends itself to confusion. This dictionary entry shows

BPE Spanish translation: a.C. GLOSSARY ENTRY English term or phrase: BPE Spanish translation: a.C. Entered by: Andrés Martínez

English to Spanish translations
Art/Literary - History / Imperio Romano
English term or phrase: BPE
Roman general Gaius Julius Caesar described the wild aurochs living in the forests of Germany in 65 BPE.

a.C. Explanation: (b.p.e. = before present era) = literalmente "antes de nuestra era". En español suele indicarse como a.C., antes de Cristo.

https://www.proz.com/kudoz/english-to-spanish/history/2014376-bpe.html

That this not just a small internet mixup is evidenced in a research article, which from an interdisciplinary perspective uses BPE & bp:

Here and following, the distinction BPE/PE is substituted for the more prevalent usage of BC/AD. For dates when the BPE/PE distinction is irrelevant, "before the present," or "bp," is used.
–– Joseph C. Miller: "Review: History and Archaeology in Africa", The Journal of Interdisciplinary History, Vol. 16, No. 2 (Autumn, 1985), pp. 291- 303.

Speaking descriptively, we know how the acronym is spelled out for sure. But we can't be sure what exactly an author might mean with the spelled out variant in an isolated sentence. An unsettled tradition of 'common use' might be as well the case as a simple mixup between BPE and BCE.

As rule of thumb: when speaking about history it is probably BPE=BCE, for whatever reason they changed an established acronym yet again;
when speaking about pre-history, as related to carbon-dating it is probably BPE=bp, for whatever reason they bloated another letter into that acronym. A weak rule of thumb, with significant areas of overlap, and strong opinions labelling the other side as incorrect. Whether BPE for archaeology itself is the correct acronym, I advise readers to ask for that in the Journal "Radiocarbon, An International Journal of Cosmogenic Isotope Research" which to this very day lists zero hits for BPE at all.

That BP as an acronym has its own set of problems seems evident. However researchers have recently focused on redefinitions different from adding letters, concentrating on meaning instead:

An official definition of the term could then be something similar to the following:

BP – A time scale which places past events in terms of “conventional radiocarbon years before AD 1950.” 1950 was chosen as the year at which to begin the BP time scale to commemorate the first publication of radiocarbon dates in December 1949. Originally, BP stood for “Before Present,” but now the symbol BP itself signifies the time scale.

The proposed meaning alteration basically codifies the way BP has come to be used in practice, with the main change being that authors would be freed from the practice of saying that “the present” is defined as 1950. It is time to institute Radiocarbon’s practice of defining BP as “conventional radiocarbon years before AD 1950” in both scientific and popular references to the time scale, finally bringing an end to “the age where the present appeared to become stuck in 1950.”
–– Jesse R Townsley: "BP: TIME FOR A CHANGE", Radiocarbon, Vol 59, Nr 1, 2017, p 177–178

Speaking proscriptively, it seems good advice to look for contextual clues when reading; and in writing to either avoid the acronym in favour of better established ones or providing a glossary explaining the intended reading for it. For either usage scenario BPE is bound to confuse readers.

"BPE" stands for "Before Present Era", and is equivalent to the more familiar BP (Before Present = 1950). It is a new "standard" abbreviation that I have seen appearing over the last decade or so (more commonly in publications from the USA).

It is worth noting that the original BP standard was established solely for the reporting of radiocarbon dates. Developments in the calibration of those dates have meant that the BP system was not used as widely as the originators had hoped (and has been used even less as the "Present" (1 January 1950) in "Before Present" increasingly became the past!).

You will sometimes find a definition for BPE provided in some glossaries of terms for recent courses, for example this 2005 course on The Archeology of Joshua Tree National Park.