Desktop PC issue - Lil help?

For what it's worth, I've seen those exact symptoms a handful of times over the years, and it was the motherboard every time. That doesn't mean it's always going to be the mobo when that symptom occurs, but since you already tried a new PSU (and I'm assuming it was a good PSU with adequate power), I can say with 99% certainty that you need a new motherboard.

So, something weird happened and I'm pretty confused.

I went ahead and ordered a new MB (went ahead and upgraded the CPU while I was at it) and sent it to my dad's house because I'm spending the weekend here. I had my computer plugged in (PSU switch on, but computer off) for a few days before I came to my dad's....The day I was leaving, I had the urge to just turn my computer on, just to see. Well, it booted all the way to the desktop, as normal. Thought that was weird, but broke it down and hauled it to my dad's house anyway. Get there, plug it in, it would boot more than the cycling on/off I was getting at first (OP), but failed before it would get to POST.

I put another MB in and am just waiting on a heatsink for the CPU to come in, but I found it really strange that, after days of sitting, this thing acted normally...until I went to my dad's hours later and it started doing the same thing (although it was getting further than it was originally).

I don't really need help, just want an explanation. I'm confident once I turn it back on with the new parts I won't have any issues, but WTF? It's almost like it stored energy while it was sitting off but plugged in, and after almost a week of being off, let me get to the desktop as normal because it had the power. Then when I went to my dad's, it's like power had drained, and would get me closer to POST, but would shutoff before it got there.