Answer: The Gemara (Shabbos 155b) teaches that while one may feed one’s own animals, one may not feed stray animals on Shabbos that don’t depend on them for food. According to Rashi, this is to avoid one performing extra tircha, bothersome acts, on Shabbos. Rambam (Shabbos 21:35), however, understands that the reason is to prevent one breaking Shabbos when preparing the food.
Thus, the Shulchan Aruch (OC 324:11) writes that one may feed one’s own animals and any other animals that depend on one for food on Shabbos. The Mishna Berura (Biur Halacha 324:14) extends this to feeding someone else's pets. One may not, however, go to the park and feed the birds.
The Magen Avraham (324:5) writes that as there is a mitzva to feed stray dogs one may feed them on Shabbos, too (See Mishna Berura 324:31; Aruch Hashulchan OC 324:2).
Based on this, the Magen Avraham, Shulchan Aruch Harav (OC 324:8), Kitzur Shulchan Aruch (87:18) and Mishna Berura (324:31) disapprove of the minhag to feed birds on parshas beshalach. Other poskim (Daas Torah OC 324:11; Aruch Hashulchan OC 324:3; Tzitz Eliezer 14:28; Shearim Metzuyanim Behalacha 87:7) justify the minhag, however.
In conclusion, even though some justify putting bread out for birds on parshas beshalach, one cannot, otherwise, feed stray birds on Shabbos.