Hopefully one of the Honda gurus here can help a VW guy out.....

My girl has a 98 Accord V6 with an auto trans. The past few days, it has been hard to start. When you turn the key, the starter solenoid pops, but the engine doesnt crank. Her dad thought it was the battery, he replaced it, still has the same problem. I did notice when it finally did start, the D4 light on the dash gear indicator lit up as well as the P. After starting and shutting down a few times, I found that running the shifter up and down through the gears would allow it to start every time. Does anybody know what could be causing this? I guess my next step is to pull the center console and check linkages and clean everything up, as im sure its nasty under there, and something may be sticking. If anyone has a solid answer for me or has had this same problem, I would really appreciate some feedback. Thanks in advance for your help.