Is the sentence "The parents then strongly requested the government to assist students stayed in the airport." correct?

Solution 1:

The full context is that there were a number of students from Hong Kong who were traveling back to Hong Kong from London's Heathrow airport, but whose flights were canceled due to a snowstorm in London. Given this, the most natural and standard way to phrase it is that they were

stranded at the airport

See e.g. here. And indeed, that was the phrasing used in a press release by the Hong Kong government at the time.

Having said all that, I should note that it is not actually wrong to use stayed. The OED does record one of its transitive meanings as

To detain, hold back, stop (a person or thing); to check or arrest the progress of, bring to a halt; to hinder from going on or going away; to keep in a fixed place or position. Now only literary.

However, while it is not wrong, it is arguably a bad choice for usage in the text of a Wikipedia article, where the context definitely isn't literary.

Discussion

The version with stayed at the airport comes from a recent (but, as of the time of my writing, revised) version of the Wikipedia article 'Hong Kong Kids phenomenon':

For instance, in 2010, the Hong Kong students could not get on the planes because of a snowstorm in London. The parents then strongly requested the government to assist students stayed in the airport.

The text on Wikipedia that is current as of the time of this writing has been changed and it reads stranded at the airport (possibly as a result the first version of this very answer).

The text on Wikipedia seems to be a summary of original text in Chinese. Unfortunately, the linked web page (in Chinese) no longer seems to have that text. Therefore, we don't know what the original Chinese text actually said.

However, there is a press release from the Hong Kong government, dated December 20, 2010, that seems to be about this event and whose first paragraph reads,

The Government is very concerned about Hong Kong students stranded at Heathrow Airport, London, due to snowstorm and would proactively offer assistance with a view to bringing them home, the Under Secretary for Security, Mr Lai Tung-kwok, said today (December 20).

Stayed

The version with stayed is not actually wrong. It used in the sense of the students who were stayed, i.e. in the passive voice. The active voice version would be [the snowstorm] stayed the students. The OED records the following meaning of the verb stay (I only reproduce the most recent examples of usage that the OED provides):

III. transitive. To stop, arrest, check.
20.
a. To detain, hold back, stop (a person or thing); to check or arrest the progress of, bring to a halt; to hinder from going on or going away; to keep in a fixed place or position. Now only literary.
^{…
1830   Ld. Tennyson Poems 96   Thou shalt not wander hence to-night, I'll stay thee with my kisses.
1873   A. Helps Some Talk about Animals & their Masters vi. 152   Among the reeds, where, at the moment,
                                                                                                                                                       we had stayed the boat.
1902   Munsey's Mag. 26 596/1   Attacking and withdrawing again before any force could be mobilized to stay him.}

However, given the setting of a Wikipedia article, this literary usage is probably not a good choice.

Choice of preposition

If one chooses to use stay, there is still the issue of what preposition should be used, since its choice depends on the verb and also on the precise meaning one wishes to convey. I have been unable to find any examples of usage of stay as a transitive verb + preposition + airport, which is not surprising given how rare this transitive usage of stay is. For what it's worth, to my ear, given the context, at seems more appropriate than in, but I am sure in is also acceptable. Even in the case of stranded, the Corpus of Contemporary American English (COCA) does record two examples of stranded in the airport. True, this should be compared to the 14 examples of stranded at the airport, but perhaps there are subtle differences in meaning. Google NGrams also shows that the version with at is about seven times more frequent than the version with at (see here).

I should also list the following examples of usage from published literature:

Actually, the point of going there this time was to hang around for a while in the airport, so I planned a Las Vegas stopover on a return trip from a speaking engagement. (source)
You must buy your ticket in the airport before you get on the shuttle train: 11 Euros one-way and 14.5 Euros roundtrip. They have an information booth in the airport that can steer you in the right direction. (source)
At one time, it seemed airport concessionaires literally banked on the fact that air travelers would pay exorbitant prices for last-minute souvenirs, gifts for kids, and even a pack of gum once they were in the airport. (source)