Whether the money is used for church purposes is irrelevant. IRS Publication 598 states:
"If an exempt organization regularly carries on a trade or business not substantially related to its exempt purpose, except that it provides funds to carry out that purpose, the organization is subject to tax on its income from that unrelated trade or business."
Fortunately, in the case of rental income from real property, such income is "excluded in computing unrelated business taxable income" (Publication 598). Caution: see content below regarding debt-financed property.
However, a second concern not addressed in the question must be raised: Will the parsonage lose its status as exempt from the real estate tax rolls of the local government?
We have worked with churches in similar situations in multiple states. Our recommendation has been to communicate openly with the local authorities (typically an Assessor) as to the church's intentions. In several cases, because the use was temporary or rented to a missionary of the church during his furlough, the Assessor did not believe its use violated the statutory exemption from real estate taxes for church property.
Readers have sometimes been helped by the following discussions:
Reader Question/Comment: Our church is looking to rent out its parsonage to a non-staff member. Can the income that is received for the rent be used for church expenses not related to the property without affecting the church's tax-exempt status?
The church tax-exempt status will not be affected using the rental income for expenses not related to the property. However, the church may lose its real estate tax exemption for the property because it is not used for an exempt purpose.
There are both potential Unrelated Business Income Tax (UBIT) and local property tax concerns. First, tax-exempt organizations generally do not need to report rental income as unrelated business income (UBI) unless it is financed with tax-exempt debt instruments.
According to IRS Publication 598, "Rents from real property...are excluded in computing unrelated business taxable income. Rents from personal property are not excluded." The IRS cites exceptions to this rule. One exception is if the rental is debt-financed, the church may owe UBIT.
We recommend that churches borrowing to finance properties from which they receive rental income should read the "Income from Debt-Financed Property" section of Publication 598.