It is in fact grammatically correct without punctuation, just somewhat awkward. Consider this analogous sentence with the same meaning, with optional parentheticals added to help you parse:

New York oxen (that) New York oxen bully (themselves) bully New York oxen.

the fact that nobody can understand without a preposition and a pronoun even after you replaced the uncommon words with common ones tells me this is a bunch a buffalo excrement that barely entertain linguist phds.

That sentence is virtually the same parse tree as "The man I saw crossed the street," which is a totally reasonable sentence to most English speakers. The difference in the parsing is that the "I" in this case corresponds to "Buffalo buffalo", a noun phrase--and the sentence is confusing because we have a tendency to add the "that" for noun phrases instead of single-word noun/pronouns, which the sentence is missing.

Crucially, what is a reasonable sentence depends heavily on words.

"History lamps traffic people house cry Michigan algebra" also has the same parse tree, but if I utter this, people will probably think I need medical attention.

For me it was confusing the first time I read it because I didn't know that "buffalo" can mean "to bully". Without that information it doesn't make sense at all.

So I never thought there's something inherently difficulty in the syntax of the sentence- I thought it's just that unexpected third use of "buffalo" that makes it so strange.

Of course, now that I think of it, what you say makes sense also. For an NL parser, the variant use of "buffalo" shouldn't be an issue. You could even say that lacking the context of what a "buffalo" really is, a parser should even find it easier to parse the sentence than a human, who must pause to try and understand what it means.

Maybe it's not such a bad sentence to test parsers after all.

'Buffalo' doesn't mean 'bully' in English. It means 'con.'

The Wikipedia article above says that though uncommon in usage, the buffalo verbs in that sentence mean "to bully, harass, or intimidate" or "to baffle".

AIUI, it's only "to bully" in the sense of verbally overwhelming someone into believing what you what them to believe. It's like the word "to snow" (in the sense that means "to con").

I would read "buffaloing" somebody as a less vulgar version of bullshitting, which is also a way of lying about something, mostly to get them to buy something they don't actually need.

I don't think "buffalo" (the verb) having a specific meaning makes any difference in how hard that sentence is to parse.