Scrum and agile says that items on the current sprint backlog should be approached in priority order, and one item at a time by the whole team.
Practically, that never seems to work for our team. Either the item is too small for all team members to be productive (including taking pairing into account). So we end up perhaps doing two or three items across the team at any one time.
I'd be interested to hear how other teams do it, and also how many items they usually commit to in a given sprint.
The advice to approach each item by the whole team is there to avoid creating mini-waterfalls within sprints, where items are passed from one specialized group to another. That leads to stuff like testers having nothing to do in first days of the sprint, then working overtime for the last couple of days when coders fiddle their thumbs. Teams should approach the problem as a whole with everyone chipping in, even outside of their respective "specialization". Yes, coders can test, testers can code and both can design architectures etc. - and in the process learn something new (amazing). That is not to say everyone should be very good at everything - it is just to say attitude like "I don't test, I'm a coder" or "I won't write this script, I'm a tester" should have no place in a Scrum team.
It is also advised to tackle items one by one inside of sprint to make sure something is actually delivered at the end. Limiting work in progress (WIP) prevents situation, where everyone did some tasks on each item, but no item has been completed by sprint's end.
However, this shouldn't be viewed as advice, not a very strict rule. For example when you have two small stories and a team of 10 it probably doesn't make sense to have all of the team swarm on just one story.
Bottom line is: no one should tell the team how to divide work among themselves, but delivering what they committed to at Sprint Planning should be expected.