Do heirs have to pay capital gains tax on property sold that was in a
living trust at the time of a parent's death?

Where do you live?  If it is the US, which State?

Also, note Important Disclaimer at bottom of page re tax advice.

Please note the disclaimer below  - no legal or professional advice.

The estate is valued as of the date of death  - or six (?) months
later -  for the calculation of estate taxes.  This new value is the
starting value for heirs' subsequent sale of property from the estate,
i.e., capital gains only apply for gains or losses subsequent to this
valuation of the property.
The "six (?) months" (not sure about the period) allows the heirs the
choice of using a lower value for property that has demonstrably
decreased in value after the death (shares, for example).

If the property was sold while the parent was alive (and still in
possession of the trstu), the capital gain will be income on thier
final tax return (filed for the year in which they passed away).  If
the property was sold after the property passed (presumably outside of
probate, if it was a proper living trust) then the tax treatment will
depend on a number of factors.

These are extremely complicated tax issues that cut across personal,
estate, and trust taxation.  I would highly recommend consulting a tax
professional on this issue.