The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system — and does so instantaneously, before any information about the measurement result could have been communicated to the other particle (assuming that information cannot travel faster than light) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the Copenhagen interpretation, the result of a spin measurement on one of the particles is a collapse into a state ...And similar paradoxes are in classical mechanics. If you are trying to predict whether an asteroid will strike the Earth using Newtonian mechanics, then you will predict the asteroid state as an ever expanding ellipsoid representing its probable locations. If you later observe the asteroid in one part of the ellipsoid, then you immediately know that it is not in another part of the ellipsoid, and you instantaneously collapse your ellipsoid.
(In fact similar paradoxes can arise even without entanglement: the position of a single particle is spread out over space, and two widely separated detectors attempting to detect the particle in two different places must instantaneously attain appropriate correlation, so that they do not both detect the particle.)
If the quantum paradox disturbs you as being spooky or nonlocal, then the asteroid paradox should disturb you the same way.
To be fair, the quantum situation is more subtle because (1) measuring a particle's position or spin necessarily disturbs it; and (2) when two particles are entangled we only know how to represent the state of the combination.
These subtleties may or may not disturb you, but they are really separate issues. The spooky action-at-a-distance paradox is fully present in the classical asteroid problem.
Books and articles on quantum mechanics seem to try to confuse more than enlighten. So to describe entanglement, they will throw in complications like the fact that an electron is not just a particle, but has wave properties.
An example of unnecessary complications is the Delayed-choice quantum eraser experiment. But this is just to confuse you, as you can see from this recent paper:
The "Delayed Choice Quantum Eraser" Neither Erases Nor DelaysIn other words, nothing to see here.
It is demonstrated that ‘quantum eraser’ (QE) experiments do not erase any information. Nor do they demonstrate retrocausation or ‘temporal nonlocality’ in their ‘delayed choice’ form, beyond standard EPR correlations. It is shown that the erroneous erasure claims arise from assuming that the improper mixed state of the signal photon physically prefers either the ‘which way’ or ‘both ways’ basis, when no such preference is warranted. The latter point is illustrated through comparison of the QE spatial state space with the spin-1/2 space of particles in the EPR-spin experiment.
Quantum mechanics is still funny because electrons and photons have both wave and particle properties. You can think of them as particles, as long as you respect the Heisenberg uncertain, and do not try to give simultaneous values for noncommuting observables. And the wave function of an electron is not really the same as the electron itself.
But many of the quantum paradoxes trick you by combining different things to obscure quantum principle involved.
By the way, it appears that the public wants NASA to focus more on those asteroid ellipsoids. Nobody really wants to go to Mars. Inhabiting Mars is just a science fiction fantasy.