Dickens seems to have had the idea for a paper or series of papers entitled 'Playing at Parliament' as far back as July 1850, when he was being exasperated by reports of the debates in the Court of Common Councils (governing body of the City of London) when many Councillors opposed sanitary reform (see Pilgrim, Vol. VI, p. 129; also Vol. 2 of [the Dent Uniform Edition of Dickens' Journalism], pp. 327–28). In one of his earliest sketches, 'Our Parish: the Election for Beadle' (Evening Chronicle, 14 July 1835; see Vol. 1 of [the Dent Uniform Edition of Dickens' Journalism], pp. 20–6), he had made fun of the self-important pomposities of local government assemblies as reflecting the absurdities of Parliamentary procedures; the Pickwick Club's behaviour in Pickwick Papers, Ch. 1, is a variant on this theme and Dickens again returns to it here. 

The name 'vestry' derives from the fact that until 1894 the parish was the basic administrative unit of the country; parishioners originally met in the church vestry for the purpose of making bye-laws, setting local tax rates, determining expenditure, etc. As the population increased, more spacious meeting-places needed to be found, but the name 'vestry' was still used to denote the assembly. 
      In para. 2 Dickens makes a facetious allusion to the famous incident related by Livy in which the Gauls are described as being temporarily overawed during their assault on Rome by the spectacle of the Senators of Rome, dressed in their robes, sitting in solemn session and awaiting their arrival.

Literary allusions

• 'Like the Senate of Ancient Rome...': see Livy, History of Rome [27–25 BCE], Book 5, Ch. 41; 
• 'Like Mr Joseph Miller's Frenchman...': refers to the compilation called Joe Miller's Jest-Book (first published 1739), which was a repository for ancient jokes such as the one about a Frenchman who drowned because of his incorrect English ('I will drown and no-one shall save me'); 
• 'new hatched to the woful time': Shakespeare, Macbeth, Act 2, Sc. 3.