As early as 1327, the English Parliament was already powerful enough to depose Edward II; was there something structurally or culturally different about England that allowed its legislative body to exert that kind of authority in what I assume would be an era of consolidating royal power in Europe?

Prepare for a long answer...

Actually, the era of consolidating power came later in the age of the absolutist monarchies in the 16th, 17th and 18th centuries. In contrast, the medieval period was a time when "Parliaments" were springing up all over Europe; for example, the Parlement of Paris, which had the power to formulate remonstrances against the king's edicts, appeared circa 1250. Yes, they often represented the interests of the aristocrats and the gentry, but kings had to work with them to get what they wanted. It was only in the 17th century under Louis XIV of France (1638-1715) that the king managed to put heavy restrictions on when the Parlement of Paris could write a remonstrance, and even ordered that the king didn't have to hear it in person to hear it. The limiting of the power of the Parlements by Louis XIV was a fundamental part of his centralisation programme that created the system we know as the "Ancien Regime", which was to be overthrown in the French Revolution.

Why did England's Parliament's powers survive where the Parlement of Paris' didn't? Firstly, I think it was because "England" was a cohesive entity long before many other European countries (such as France), enabling the Parliament that was established by Simon de Montfort in 1265 to remain largely unchanged in the areas it represented until the various Acts of Union with Scotland and Ireland in the 18th and 19th centuries.

Secondly, the first English Parliament was created against the backdrop of the Barons Wars of the 13th century. One of the outcomes of the First Barons War (1215-1217) between King John and his Barons was Magna Carta, which limited the king's ability to illegally arrest people and to take feudal dues from his barons. In the Second Barons War (1264-1267), Henry III (John's son) was opposed by his barons due to raising high taxes in a time of famine. They wanted to reissue Magna Carta, and institute a Parliament made up of the leading men of the realm, which would, in future, be need to grant the king permission to raise taxes. Although the baron's leader, Simon de Montfort, would be killed in battle, future kings were forced to call Parliament and get their support every time they wanted to raise taxes. In some way, this ultimately helped English kings, as they could be seen to have the support of the country when taxes were confirmed by Parliament. I am not so familiar with the origins of the French Parlements, but the fact they could only react to what a king did, instead of being vital to the process of tax collection, means that the English Parliament was much more difficult to brush aside. The English Parliament would only be strengthened during the 16th century, when the Tudor monarchs used Parliament to push through the religious changes that constituted the English Reformation, and give them the sheen of public support. Ultimately, the Tudors using Parliament in this way led to the Civil War, as Parliament felt it had the right to have an input on issues other than taxation.

So, back to the point that the English Parliament was an important way for the King to be seen to have the support of the country. In 1327, when Edward II was deposed, it was not Parliament that pushed for the deposition of the king, as they were to do in the 17th century during the English Civil War, but Queen Isabella and the leading magnate Roger Mortimer. As an English king had not been deposed before, they needed Parliament to make it appear that the king's deposition had public support (which it probably did, given how badly Edward had managed the country).

So, in conclusion, I would argue that although the Parliament of 1327 was not nearly so powerful as the one that had Charles I executed in 1649, it was still vital for the king so his taxation policies could be seen to have popular support. As Edward II had been so utterly defeated by Isabella and Mortimer, they used Parliament in the way the king would have to give the impression of popular support for Edward's deposition. Parliament was not self-conscious yet, but the fact it was created during the Second Barons War meant that it had more power than its continental counterparts, because it provided a service for both King and the barons.

tl;dr Although the English Parliament limited the king's powers, in that it was needed to raise taxes, it also gave the king's policies the air of popular support. Consequently, kings were happy to use it. When Edward II was defeated by Isabella and Mortimer, they acted as the king and used Parliament to bolster their legitimacy. Edward's deposition did not come from Parliament itself.