Okay...I'm going to 'um, actually' this post, but the tl;dr is no, this is not real, and hopefully that won't change how you feel about these amazing creatures because they truly are incredible.
So no, jumping spiders do not raise their young. Other than a species of ant mimic jumping spider, which is far less "cute" than the species here, this is just not a behavior of this family of spider. Like most spiders, jumpers will care for the egg sac, but once it hatches, the young (called slings) are self-sufficient and will leave the nest immediately. Additionally, a single egg sac can produce anywhere from 50-200 eggs, so raising offspring would look very different than these pictures (most of which appear to be photoshopped, staged, or displaying the sexual dimorphism between male and female of the species--the males tending to be significantly smaller).
However, if you DO want to see something cool, please consider checking out the lycosidae family of spider, which is the family of wolf spiders. These spiders DO exhibit more paternal behaviors; the female of many lycosidae species will first carry her egg sac under her abdomen, and will then carry her young on her back until they are large enough to hop off and take care of themselves. I won't attach pictures because while I think they're adorable, wolf spiders don't tend to get the "cuteness" pass that jumpers do. If you're not afraid of spiders or if you're willing to try it out though, I highly recommend looking this up--it's such a cool behavior and I frankly think wolf spiders look like puppies (it's something about the eye configuration).
THAT SAID. Just because Jumping spiders don't raise a single offspring to maturity as this post seems to be trying to suggest, don't let that change your opinion of these incredible creatures. Jumping spiders are amazingly intelligent and curious animals, and are commonly kept as pets. They exhibit distinct personalities and many individuals can be acclimated to being handled. I've even had several wild jumpers voluntarily walk onto my hand and chill with me (it's super adorable when they're curious about you, too--they will LITERALLY do the puppy head tilt while regarding you). They're also incredible hunters, using ambush techniques rather than webs to capture prey, hence the amazing jumping ability.
This kind of anthropomorphizing of creatures typically seen as "creepy" is pretty disingenuous, because it implies animals have to behave the way humans do before we see them as worthy of respect. Spiders are incredible animals who get a very bad rap for very little reason, but if you take the time to learn about them, they are absolute marvels of evolution and survival that have been wildly successful in filling a massive variety of ecological niches. I encourage anyone who is interested in them (or even slightly intrigued) to watch some videos or read some articles because there are truly awesome things to learn!
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