Why do you want to put them in parallel? Series strings are perfectly good and reduce the overall current needed, as well as needing only one current-limiting/balancing resistor per string. Think about it: if each LED needs, say, 20mA then 7 in parallel would need 140mA whereas 7 in series would need only 20mA per string. 7 LEDs per string, as you had previously, was fine and would need a supply voltage of around 15V minimum (depending on the type of LED). What type/colour of LED do you plan to use?
Yes, in a parallel arrangement.