From the MM.
Variant Familiar: - pg 63
The quasit can serve another creature as a familiar, forming a telepathic bond with its willing master. While the two are bonded, the master can sense what the quasit senses as long as they are within 1 mile of each other. While the quasit is within 10 feet of its master, the master shares the quasit's Magic Resistance trait. At any time and for any reason, the quasit can end its service as a familiar, ending the telepathic bond.
The imp and pseudodragon have similar options.
VARIANT: FAMILIARS - pg 347
Any spellcaster that can cast the find familiar spell (such as an archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell (see the Player's Handbook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.
So if npc's can do it, why not player characters.
Now the answer to the above question comes down to if it will impede the enjoyment of another player, so if you have a chain pact warlock in the party you might want to not just let the wizard in the party have a super cool familiar without it costing him something.
On chain pact warlocks, when they cast the spell find familiar there is no choice for the creature it is bound to him and can't just leave. Also it never mentions they would get the magic resistance abilities like a willing master/familiar relationship would have. The find familiar spell even says you have fiendish, fey, or celestial versions of the creatures suggesting that a warlocks pseudodragon might not be a real pseudodragon at all but could be an angelic copy of one.