I'm trying to explain the basics of Bitcoin to my parents.
If this is your objective, I would argue that it is not necessary to go in-depth into asymmetric encryption/signing. The basic mechanics of Bitcoin, or blockchain systems in general, are easily, (and probably best) explained on a very abstract level, i.e. it is a decentralised solution to asset transfers and consensus building, and so forth.
As for the signatures, you could say that they serve as transaction authorisation so that only the owner of tokens is able to spend, but everyone can verify the signature.
What's an extremely simple asymmetric cipher I can use as an example?
If for some reason, your audience is really interested in learning how it works on a more technical level, I would avoid RSA and the effort of doing the transfer from encryption to digital signatures later. In this case, you might simply want to stick to DSA (the classical, non-elliptic curve version), and use small numbers to talk them through step by step.
Not knowing your parents/friends at the pub, I would have my doubts though that they care about the hardness of discrete logarithm problems or finite group operations, and from my experience with similar settings, I advise you to start with the general concepts, and then go into the exact algorithms if required.