Yes, You may take up the transfer in the Second year of articleship if your parents have shifted from one city to another city. If you are willing to work with the same firm in its branch office, i.e. in Delhi, then its perfect but if you don't wish to join there, then in that case you may contact ICAI reporting them with a better reason for not working any longer with the same firm at its Branch office. This will resolve your problem.

Goodluck 

you can transfer within 1 year of  starting your Article ship trainig.  You have valid Reasion so there must be provision for Transfer. Do one thing raise your Query on e-sahhayta.

Find Direct Link OF e-sahhayata(ICAI) :

https://220.227.168.109/ 

Thanks alot for your replies.

I have already posted my query but did not receive any verification mail...

you can take termination . it is valid reason 

Cheak your Status of Query @ e-sahhayata.