Why does Charles Dickens draw a parallel between Paris and London in A Tale of Two Cities? How are the French characters depicted in the novel compared to the British characters?

I am going to answer your second question first. Charles Dickens did create a huge contrast between his French characters and the English, depicting most of his French characters as much more volatile and violent than their English counterparts. Compare Charles Darnay with his uncle, the Marquis St. Evrémonde. Evrémonde is haughty, arrogant, and heartless, the perfect example of French aristocracy. Darnay's blood is no less aristocratic, but his character is far more noble, kind, socially conscious, and concerned with justice. These are traits he shares with several of his English compatriots. Even the irredeemable Sydney Carton possesses more empathy and chivalry than the "noble" Evrémonde. This dichotomy presents in several character pairs: Miss Pross and Mme Defarge, Mr. Lorry and M. Defarge, Jerry Cruncher and John Barsad, and so on.
Now the question is why. Why juxtapose French volatility so glaringly against English stability?
Meltem Kiran-Raw, a Lecturer in English, Department of American Literature and Culture at Baskent University, explains:

In the eighteen-fifties, Charles Dickens was concerned that social problems in England, particularly those relating to the condition of the poor, might provoke a mass reaction on the scale of the French Revolution. In a letter written in 1855, for example, he refers to the unrest of the time as follows:
I believe the discontent to be so much the worse for smouldering, instead of blazing openly, that it is extremely like the general mind of France before the breaking out of the first Revolution, and is in danger of being turned . . . into such a devil of a conflagration as never has been beheld since. (Collins, Irene. "Charles Dickens and the French Revolution." Literature and History 1.1 [1990]. Page 42)

Kiran-Raw suggests that A Tale of Two Cities was further social commentary on Dickens's concern. Even in the novel, she maintains, he editorializes on his present-day England:

"The period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, in the superlative degree of comparison only" (1; bk. 1, ch. 1).

Other researchers at Stanford write that Dickens researched carefully for his novel and wrote a highly accurate summary of revolutionary France (dickens.stanford.edu), possibly to highlight that history was in real danger of repeating itself. However, by presenting his English characters in such an admirable light, I believe Dickens also gave his readers hope and encouragement that their society could rise above the social concerns of the day.
http://dickens.stanford.edu/tale/historical_context.html

http://www.victorianweb.org/authors/dickens/2cities/turlit12.html