Does the same current pass through each bulb in a parallel circuit?
No, the magnitude of the current going through each bulb would be, at least slightly, different.
To keep the discussion simple, assume we are talking about 2 bulbs wired in parallel. Even if both bulbs came from the same package of bulbs (therefore same wattage) there would be a slight difference between the results of a careful measurement of resistance of the 2 bulbs. They would be within a few % of the same resistance, but not 100% identical. There would be no reason for the manufacturer to provide 100% consistent resistance values.
Ohm's Law says that
Perhaps by "same current" you mean the actual same electrons. When the electrons in the wire come to the point where the 2 parallel paths separate, each electron has to decide which path to take. In a parallel circuit no electron goes through both bulbs.
If the 2 bulbs were wired in series instead, then the current in the 2 bulbs would not have a choice. Each electron must pass through both (unless a few of them do not make it to the 2nd bulb before someone turns the switch off).
I hope this helps,