Increase brightness of parallel LEDs
Unless there is a hidden resistor which may be in line inside the battery compartment or by the first led,  the string depends on three things. 


Matched diodes with close if curves. 
Less important, the number of diodes in parallel causing a voltage drop. 
MAINLY that the batteries should be lower than the led VF for a decent current.


Ways to fix this:


Add appropriate resistor for n Leds in parallel. Since we don't know the VF or IF, would could assume 3.4V at 20mA. I'd hesitate to size it at 500 mA due to potential runaway issues (one led draws more, blows out, allowing all to draw more, then next one blows, rinse repeat). So aim for 100 mA total, meaning 4 mA each if they are all equal ((Vs - Vf) / 0.1A). Risky due to unknowns.
Rewire all the leds for their own resistor at 10~18 mA. Part heavy.
Rewire for multiple parallel strings of series leds, basically like typical led strings. Depends on how high your power supply goes. If 12V, then it's three in series plus an appropriate resistor.