Trouble starting my 2004 Lexus RX 330

My Lexus RX330 has been having difficulty starting over the past year and a half. I used to be able to get it to start by letting it roll backward and trying to start it again but now nothing seems to help and its becoming more frequent. When this occurs the check engine light, battery, oil and A/T oil temp light turn on. I replaced the battery, starter, and some old cords. If you have any idea please please let me know

Time to check other parts of the starter circuit. The interlock that’s part of the gearshift would be where I’d start. Does putting it in ‘N’ instead of ‘P’ (with your foot on the brake, of course) have any effect?

Next might be the ignition switch - has jiggling the key ever helped?

A good independent should be able to track this down.