If a person has a YouTube channel where that person displays their original music, and the channel is monetized (that is, the channel owner is making money from ads), is the channel owner allowed to put pictures in the videos from sites like Pixabay, Flickr, etc?
As far as copyright goes...
Pixabay: Yes (with minor exceptions)
Pixabay's license is quite broad, and allows for unattributed commercial use with a small number of exceptions. You can't "use images with identifiable brands to create a misleading association with a product or service" or "portray identifiable people in a bad light or in a way that is offensive," but otherwise it would allow for use in a video (there are other exceptions, but they wouldn't really apply to such use).
Flickr: It depends on the license
By default, images are not licensed for additional use other than viewing on Flickr. However, users are free to license their images as they wish, and Flickr explicitly supports various Creative Commons Licenses, many (but not all—particularly the NC licenses) of which would support such commercial use. Depending on the license, additional requirements, such as providing attribution, may apply.
Note that there are non-copyright concerns that might come up in specific situations, such as trademarks (I mentioned one potential issue in my explanation of the Pixabay license) or model releases. Another answer to this question has some good information on these concerns.
It Depends, but Probably Not
You may not make copies of someone else's copyright images without permission. Even more, you may not use such images to help you make money without permission.
Sites such as Flickr list the available license or permissions for each posted image. Some images will be free for anyone to use in any way. Some will allow use only under specified conditions. If there is no statement of permission, then the image may not be used in any way at all, unless an exception to copyright protection, such as fair use or fair dealing applies. But using the whole image for a commercial purpose is unlikely to qualify as fair use. So what you may do will depend on the particular image and the license specified for it. But images that allow reuse for profit are less common than ones allowing narrower use.
If you use an image without permission, the copyright holder could sue and win damages. But more likely, and more economically for the copyright holder, a complaint could be made to YouTube. YouTube has policies under which infringing content can be taken down without warning, ad payments can be diverted to the copyright owner, and repeat infringers can lose their YouTube accounts. These decisions are made by YouTube in accordance with its ToS, and no court process is needed. Nor do they have to grant any hearing to an alleged infringer.
As to fair use, many claims of fair use are made online that would never hold up in court, but copyright owners don't bother to sue because there isn't enough money at stake. Also, fair use applies only if the suit is in the US. Fair Dealing, which applies in the UK and some other places, is significantly narrower.
Don't risk it
I am (or have been) a semi-professional photographer, so I have some background information that I think you might find useful. In general, it's even riskier than David Siegel says in his excellent answer. Just because you have permission to use an image from a photographer or website, doesn't mean that the photographer or website has the correct authority to give you permission.
For example, if a photographer takes a great shot of a can of Coke, and gives you permission to use the image, you can still be sued by The Coca-Cola Company for using their brand without permission.
I found this shot of a man on Pixabay (it's safe for work). if the photographer didn't get him to sign a model release form, then that man could one day sue that photographer for publishing that image. But, he doesn't have to -- he could bypass the photographer and go after the rich you-tuber who used that image to generate revenue instead. That's why I've linked to the shot rather than downloading and embedding it here. Even if it were covered by fair use, I couldn't afford to fight that man's lawyers if he chose to come after me, even unfairly.
So, even if you have permission to use the photograph, that's not the only permission you'll need. To get around this, you should only use reputable, professional, stock image sites, like Getty Images (with which I am not affiliated), who properly vet their photographers and their images and insist on getting releases from all models and subjects before publishing the image as part of their catalogue.