I've often wondered this. Fortunately, there's a study:
Endocrine Journal Vol. 55 (2008) , No. 3 613-616
36 post-menopausal women with established osteoporosis received teriparatide 20 mcg daily for 18 months and then were followed for another 6 months. You can see that iPTH significantly declines for months 6-18. There was actually no significant increase in calcium.
The authors mention this as a potential way to monitor compliance. I have wondered if teriparatide could be used to treat osteoporosis in patients with mild asymptomatic primary hyperparathyroidism or even secondary hyperparathyroidism. I had assumed that the PTH would be additive and risk driving calcium higher. In fact it may possibly lower PTH in these conditions. That would be an interesting study...