Yes, most of great science fiction parallels problems of the present. But not specific elements of the present. Not specific people who are alive in present day or specific organizations.
That depends on the particular work. A lot of characters in classic SF have been allegories for the Communists or the Nazis or Joseph McCarthy or what-have-you.
I watched a few first season episodes and my feeling is that the premises for the episodes are awesome but the jokes aren't funny. I think the reason I like the newer episodes better is that they focus on letting the jokes flow from the characters based on science fiction tropes. Not just making specific references to specific things in present day.
That's not an early/late divide. Just about any era of Futurama
has a mix of stories that use science fiction ideas to drive a character-based narrative (e.g. "Parasites Lost," "Time Keeps on Slippin'," or "The Sting") and stories that are just riffs on present-day trends with vague trappings of future technology (like "Raging Bender," "Bender Should Not Be Allowed on TV," or "Attack of the Killer App").
You say you've only seen the first and last seasons so far. Well, the last season has a lot of the strengths that developed after the first season, but those strengths arose far sooner, and the last season is not as strong overall as the intermediate seasons were.