The way I understand, it has become common in recent times for the parents of the girl, to offer a monthly stipend for a period of time after marriage. Is there a source for doing this?
This practice, or at least the idea broadly, is endorsed by Rambam who writes the following in Hil. Matnot Aniyim (10:15):
הנותן מזונות לבניו ולבנותיו הגדולים שאינו חייב במזונותן, כדי ללמד הזכרים תורה, ולהנהיג הבנות בדרך ישרה ולא יהיו מבוזות, וכן הנותן מזונות לאביו ולאימו--הרי זה בכלל הצדקה; וצדקה גדולה היא, שהקרוב קודם
One who provides sustenance to his grown sons and daughters whom he is not required to support so that the males learn Torah, and so to lead the daughters in the proper path and that they not be disgraced, as so too one who gives sustenance to his father and his mother--this is included in charity. And it is a significant [example of] charity; for relatives take precedence. (translation my own).
Significantly, his wording ולהנהיג הבנות בדרך ישרה ולא יהיו מבוזות definitely seems to include cases besides for supporting the husband learning, and apparently cases besides for supporting the wife learning. Furthermore, his natural transition to speaking of supporting one's mother and father, making no mention of doing this to facilitate their study, strongly implies that besides for support to facilitate study, Rambam endorses all support for grown family members who could use it. Today, many people's primary financial concerns begin after marriage, and it is then that Rambam's statement would be most applicable.
Indeed, Rambam writes explicitly in Hilkhot Matnot Aniyim (7:13) that supporting one's family takes precedence over charity to others:
עני שהוא קרובו, קודם לכל אדם
Note: Rambam does not limit his statement to support after marriage. It seems probable that continuing to feed and house one's unmarried children would qualify as well. However, he certainly includes cases after marriage.