A sensitive parity mixing measurement aimed at improving the precision on the value of the Weinberg angle at low momentum transfer and determinations of parity-violating nuclear anapole moments will be performed. The experiments will exploit the predicted large parity mixing effects in atomic ytterbium. Measurements of parity mixing for a series of isotopes will make it possible to essentially eliminate atomic theory uncertainty in extracting the Weinberg angle, which will be due to lack of knowledge of changes in nuclear neutron distributions from isotope to isotope.