Is there any way I can run 32-bit software that was designed for earlier Macs on my new 64-bit Mac?

There's no practical way to do this.

All of the versions of macOS that can run 32-bit Intel applications are Intel-only themselves. Virtual machines run the same basic type of machine code as the host does. So on an Apple-Silicon-based Mac, like yours, you cannot run any pre-Catalina version of macOS in a Parallels Desktop or VMware Fusion Pro virtual machine.

What you would need is a program that emulated an Intel CPU and all of the hardware of a Mac built around an Intel CPU. Emulators are much, much slower than virtual machines – and I don't believe that any of the emulators which might be available for the Mac (e.g., UTM or QEMU) are up to the job of running old versions of macOS at all.

Well, there's a company called "Passport Music Software" that claims to be bringing out a new version 6 of Encore which is 64 bit "this summer". And claims to be "Brought to you by the original creator, Don Williams".

External link: https://passportmusic.com

There is an "about us" section on that site with more details. And a discussion board, which seems active.

Disclaimer: I'm in no way associated with this company, and cannot provide you with any information beyond what is on that website.

Other alternatives:

You could consider getting an old Intel-based Mac that can run Mojave, to serve as a backup should your 2014 MBP stop working. (If Encore is encumbered by activation-style DRM, and the vendor has shut off the activation servers, this might not be an option.)

Buying a used Mac can be risky, especially if the previous owner has not properly prepared it for a new owner – but Other World Computing has a number of old Macs for sale, and might be a fairly safe source.

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You could also start looking for a replacement application that runs on current machines and does the same sorts of things that Encore did. The big issue here, besides finding equivalent functionality, would be whether there was any easy way to migrate your data, so you didn't have to redo all of the work on your 500+ songs from scratch.

I don't know what applications are available, or which offer the best migration paths, but maybe other Mac users in online music communities might.

Mojave was the last macOS to run 32-bit applications.

With the release of macOS 10.15 Catalina and forward, applications will have to be 64-bit. App developers will issue updated apps if they want their apps to work on this newest macOS upgrade

32-bit app compatibility with macOS... - Apple Support

32-bit app compatibility with macOS High Sierra 10.13.4 and later - Apple Support

Mudge wrote:

There is an "about us" section on that site with more details. And a discussion board, which seems active.

Not just that. While the "About Us" section has a bit of history about various people, products, and companies, the "Development" section shows the types of updates they plan to make to Encore 6, relative to Encore 5.

MacTracker indicates that the 13” and 15” 2019 MacBook Pros originally shipped with macOS 10.14.5 (Mojave), and so can run that or 10.14.6 (the final release).

The 16” 2019 MacBook Pro originally came with mscOS 1015.1 (Catalina), and so cannot run Mojave.