Upon encountering a parsing error, bc(1) passes an error message on to
dc(1), which writes the error message to standard output along with the
normal output.

That is a bug. Error messages should go to standard error instead, as
POSIX specifies:
http://pubs.opengroup.org/onlinepubs/9699919799/utilities/bc.html#tag_20_09_10

GNU 'bc', Solaris 'bc', and (as of 2017) OpenBSD 'bc' act like POSIX says and write error messages to standard error.

Bizarrely, the exit status of bc(1) is left unspecified:
http://pubs.opengroup.org/onlinepubs/9699919799/utilities/bc.html#tag_20_09_18
And indeed, all versions of 'bc' exit with status 0 if there is an input
error such as a parsing error, so the exit status cannot be used to
catch it. That leaves examining standard error as the only method for a
program calling bc(1), such as a shell script, to distinguish between an
error state and normal operation. That is, with this bug, there is no
way at all.

The following example shell function transparently hardens bc(1) by
intercepting standard error and exiting the program or subshell if an
error was produced.

bc() {
	_bc_err=$(command -p bc "$@" 2>&1 1>&3)
	[ -z "${_bc_err}" ] && return
	printf '%s\n' "$0: bc(1) caught errors:" "${_bc_err}" 1>&2
	exit 127
} 3>&1

This bug was fixed on OpenBSD 'bc' back in Feburary 2017. I figured FreeBSD would eventually pull in that fix since FreeBSD 'bc' is based on OpenBSD 'bc', but as of FreeBSD 12.0 that hasn't happened.

The OpenBSD patch that fixes the bug is here:
https://marc.info/?l=openbsd-tech&m=148767278323276&w=2