Think about any story or character you like. You probably see some kind of meaning in them which makes you like this. Not moral lessons or advice, like "lying bad" or "discipline good", but some kind of interpretation of the character that comes to mind which makes you like the story or character. I will give you an example, I like Kiryu Kazuma from the Yakuza games. I see him as a man who has both impressive physical skill and intellect, but with a heroic morality and humbleness due to which he prefers isolation and avoids luxury as much as possible. Due to his humble nature, he manages to avoid trouble unless a truly despicable villain finds him, in which case he goes on a violent crusade that is completely justified. I like Kiryu Kazuma because I see these things as admirable qualities.
But, this is all just an interpretation. In fact, even if you get descriptions of Kiryu's character within Yakuza games, even those are only interpretations of that character or the writer, not objective truth. But is it possible to get an objective true analysis of Kiryu's character? If you try, you'll notice that any interpretation of a narrative or a character depends on multiple assumptions you have made irrationally, either by instinct or indoctrination. If you try to avoid any irrational assumptions, it becomes impossible to make any interpretation at all, it just devolves into a set of facts with no way to connect them. You may have noticed how you and another person can watch the same movie or play the same game, so in essence you may both know the same set of facts, but you make like a character and the other person may despise the character. And neither of you can convince the other because you keep arguing about the facts, when the real reason both of you have those different interpretations and reactions is because of those irrational assumptions. Going back to my interpretation of Kiryu Kazuma, some I've noticed are "Crusades must be justified by you being completely in the right", "A humble nature is superior because avoiding trouble is better than looking for conflict", "Even if a person has great ability and can be very succesful, it is heroic to still live in isolation and relative poverty than living like a king". Due to these irrational assumptions, your interpretation may even be in contradiction with other facts in the narrative.
Deconstruction means examining the irrational assumptions that you can find in your own interpretation or reaction towards a narrative or character, or looking at the irrational assumptions that the public interpretation of a character is built upon. When you engage in this, you may find that you have assumptions that you don't really agree with but had been applying without knowing, or you may find that when you remove one or more of these foundational assumptions, the view of the character changes. This isn't an exercise where you change those assumptions, it's just about recognizing them and seeing how they apply to other parts of your life or finding out why you have them at all.
When media is described as a deconstruction of a character or a narrative, it means that that media was created to reveal the irrational assumptions behind the popular perception of that character or narrative, or to show the writer's deconstruction of their own view of that character. The later Yakuza games engage in a deconstruction of Kiryu Kazuma's character this way, like pointing out how Kiryu's humble and avoidant nature may also just be an ultimate cowardice, because he refuses to stay after his victory and cause greater change, instead he just runs away to hide again and puts other people in charge, over and over again. It's not just about using logic to criticize, it's just an examination of why people like it and explaining the contradictions caused by those assumptions. A spoof can be a deconstruction, but this is rarely the case, because a parodies usually depend on changing the circumstances around a character and having that character repeat the same action regardless. (Example: Austin powers - What if james bond was ugly but still acted like a suave man? This isn't a deconstruction, this is just a joke.) I haven't seen Archer so I can't comment on the examples above, but I hope I have explained the concept well enough to make the difference clear. An example of both parody and deconstruction done well in a story is Cobra Kai.
Most people who call their work deconstruction are engaging in this intellectual exercise, usually because they aren't smart enough to mentally simulate a hypothetical mind and remove foundations from an interpretation and watching the changes. Also, to deconstruct something you need a detailed view and an interest in what you're criticizing to do it at all, since you need a good knowledge of the facts to understand how an interpretation would change, as there would be otherwise ignored facts you need to bring back into context again. Sorry for longpost, I've tried my best to explain.
