I’m confused about what you do. Are you a designer, a printer, or a little of both?
I’m not an attorney, but here’s what I understand to be the case in the U.S.
In the U.S., the copyright stays with the designer unless (1) the designer is an employee of a company and that employee produces the work for the employer or (2) the independent designer signs over the copyright to the client. In the latter’s case, most clients don’t realize that they don’t own the work produced by the designer and that, in the absence of a contract stating otherwise, they only have purchased rights to use the work.
If the client supplied the images or fonts in the work, the client either owns the copyright, has obtained rights to use the images, or has illegally obtained the images. Without a contract stating otherwise, you can’t legally use those images. However, if the images in the work are yours or obtained under your name through a stock service, you own the license to use those images for your purposes (depending on your licensing agreement with the stock service).
When your role is printing the artwork supplied to you by a customer, you don’t have any rights to use that artwork without their permission.
Of course, there’s more to consider than what the laws say.
I doubt any client would care if you posted their no parking sign on your website, but you never know. If the no parking sign also contains their logo, they might care since it could imply an endorsement they don’t wish to give. Even when you own reproduction rights as the designer, do you want to offend a customer by posting the work without their permission? After all, your brand and reputation are at stake. Worse still, do you want to receive a threatening letter from an attorney, even when a court battle would rule in your favor? Win or lose, it’s best to avoid litigation.
Even so, as @Smurf2 said, sometimes it’s better to ask forgiveness than permission, but if it were me, I’d be cautious. Most clients will readily give permission, so when possible, get permission.
As @PrintDriver and @Mojo said, it’s best to explicitly state in your contract that you have the right to use the work for self-promotional purposes. Most clients/customers won’t object, and for those that do, you can scratch that clause from the contract.
As I mentioned, I’m not an attorney, so take what I say with a grain of salt. If you want solid legal advice instead of the opinions of a guy on the internet, ask an attorney.