Pay Our Coast Guard Parity Act of 2023 This bill provides continuing appropriations to the Coast Guard for pay and benefits when there is a Coast Guard-specific funding lapse. Under the bill, a Coast Guard-specific funding lapse occurs when (1) a bill providing appropriations for the Coast Guard for a fiscal year has not been enacted before the beginning of that fiscal year, and no joint resolution providing continuing appropriations for the Coast Guard is in effect; and (2) a bill providing appropriations for the Department of Defense (DOD) for the fiscal year has been enacted before the beginning of the fiscal year, or a joint resolution providing continuing appropriation for DOD is in effect. If a Coast Guard-specific funding lapse occurs, the bill provides appropriations to the Coast Guard for pay and allowances for military members of the Coast Guard who perform active service or inactive-duty training; pay and benefits for certain civilian employees and contract employees; the payment of a death gratuity; payments for travel related to funerals, the dignified transfer of remains, and unit memorial services; and the temporary continuation of the basic allowance for housing for dependents of members of the Coast Guard dying on active duty. The bill generally provides the appropriations to the Coast Guard until the earlier of the enactment of specified Coast Guard appropriations legislation, the termination of the availability of appropriations for DOD, or two weeks after the beginning of the Coast Guard-specific funding lapse.