I am currently playing a bass clarinet in my concert band, but my parents want me to switch to a "lighter" instrument. I looked through different options, and saw the soprano saxophone. If I were to make the switch, how hard would it be (Embouchure, fingering, etc.)?
Sax and clarinet are extremely similar in both embouchure and fingering. There would be a tremendous amount of skill transfer if you were to switch.
That being said, soprano is sort of a side member of the saxophone family. Lots of music that calls for saxophones doesn't call for soprano. It would be more practical (and cheaper) to start on alto.
my parents want me to switch to a "lighter" instrument
This is an awfully strange reason to want to switch. What's really going on here?
It's not a trivial change, but it's probably one of the easier changes. The embouchure is obviously narrower, more akin to Bb clarinet. The fingerings are related, but sax fingering is somewhat easier anyway. You'll need to practice, but it will be easier than switching to, say, euphonium.
Be aware that soprano sax is a bit more specialised than alto, tenor or bari. There may not always be parts for you. You'll need to discuss this with your band director. I've played in a band with a soprano before, and half the time he had to play tenor parts. Also, sopranos can be challenging to play in tune.
If you really want to switch, I'd suggest alto sax or Bb clarinet before soprano sax.