Yes. And if you'd like to know what those periods have been, historically, the link I provided is useful for educating yourself.
In any particular instance, certainly not. However, at some point "coincidence" starts to become an unsatisfying explanation.
Here's a way to think about it. Let's say you flip a coin and it comes up "heads." Will you think it's weighted towards heads on that basis? No. It could be random. Now you flip it again, and it comes up "heads" again. Still could be coincidence. Same with the third "heads." But if it were to come up "heads" nine times in a row, the probability of that happening randomly is less than 1/5 of 1%. So, it makes sense to think maybe it's not random, at that point. Maybe the coin is weighted.
In the same sense, if you have a recession start under a Republican president, that could be random. But what if a recession starts under each of the last nine Republican presidents? What if, in fact, sometimes two or even three recessions start under some of those Republican presidents? Well, you might say, recessions start under every president, so that doesn't mean anything. Only that's not true. No recession started under Obama. Or Clinton. Or Johnson. Or Kennedy.
Is it just a big coincidence that we fall into recession so often when Republicans are in the White House, and so seldom when Democrats are? Perhaps. It's impossible to say with what is, by necessity, a fairly small data set. But the dynamic I talked about in my post would be consistent with the pattern I'm seeing -- if, basically, Republicans tend to take control when Democrats have created a strong economy, because more voters are thinking of themselves as future millionaires who will benefit from Republican policies. If that's what's happening, then the tendency would be for Republicans to take charge when growth cycles were already pretty old, so there was a heightened risk of recession. Meanwhile, Democrats would tend to take charge either in the midst of a recession that started under a Republican (like Obama) or shortly after one ended (like with Clinton). So, they'd tend to take charge when the growth cycles hadn't even started, or were quite young, allowing for longer run-ups. That would go a long way towards explaining the pattern. It wouldn't, however, do much to explain why we tend to get multiple recessions under Republican presidents (e.g., the two that started under Bush, or the two during the eight years of Nixon/Ford, or the three that started under Eisenhower).