What regression to use to calculate the result of election in a multiparty system?
Robert is right, multinomial logistic regression is the best tool to use. Although you would need to have a integer value representing the party as the dependent variable, for example:

1= Conservative majority, 2= Labour majority, 3= Liberal majority....(and so on)

You can perform this in R using the nnet package. Here is a good place to quickly run through how to use it.