1997 Mercury Marquis Cold Hard to start

The engine will crank and run slighty then just crank. This is about only 3 seconds. So far the parts replaced are the battery, starter, fuel pump and filter. If I turn the key on and wait till the fuel pump stops and then turn it off and repeat this step ten times, then it will give it a little boost and run about 4 seconds. After about 5 to ten tries the engine will finaly start. From there its fine for the rest of the day. The engine runs like new and after it is warmed up. Could the fuel have something to do with this? It gets the cheap stuff. Posibly a senser switch or a relay? Im an old school mechanic and this stuff you just replace parts. Well I hope you can give me some help on this. My Parents just got this car from the original owner who kept it in a garage but had the same problem. Its the Grand Marquis 4.6 SFI and looks just like the new ones.