... as the companies appear set to undergo U.S. antitrust probes.
"set to" means "prepared to" or "ready to" here, but with a strong implication that they are prepared because it is going to happen. So in effect, "set to" means "highly likely to".
Consider a parallel construction:
Jane appears ready to run the race.
I believe "ready" is a bare initiative form here, and "to run the race" is the object of "ready". Or perhaps it would be better to say that "appears ready" jointly form a phrasal verb, with "to run the race" as its object, and "Jane" as the subject. In that case, "appear set" is also a phrasal verb, with subject "the companies" and object "to undergo U.S. antitrust probes."
... agreed to split up oversight of technology giants
This means that the US Federal Trade Commission (FTC) and the US Department of Justice (DOJ) have agreed to divide the task. According to the linked article:
... the FTC will oversee antitrust scrutiny into whether Facebook’s practices harm competition in the digital market. Reuters reported that the DOJ has been given jurisdiction over a potential probe of Apple.
So the split is between The FTC and the DOJ, not a prospective division of the companies into smaller parts.
"Google parent" is "headlinese" or "journalese". "Google's parent " would be more strictly correct, but in a news story, particularly in a headline or in an opening paragraph (or "lede" as it is known) brevity is of great value, and so two characters and one syllable are saved. This has been used often enough that the form "X parent, Y" for "Y, the parent of X" is in common use, especially in financial journalism. Often the parent company is the one legally involved in some action, but the subsidiary is far better known, as is the case here: Google is much better known than Alphabet. The form "X parent, Y" allows the emphasis to be put on X, the well-known name.
The question did not ask about it, but some may be confused by the use of "tumbled" in the opening sentence of the article. This means that the stock price fell suddenly and significantly.