Quantitative easing and interest rate parity Suppose a country initiates quantitative easing by printing money and buying government debt. This will put a downward pressure on interest rates. Will this action tend to depreciate the country's currency against USD because money supply has increased decreasing the value of money? Suppose, further, that the final interest rate after quantitative easing is less than the USD interest rate. From interest rate parity, I believe it would follow that the currency would appreciate against the dollar. If yes, there are two opposing influences on the currency. Which one will prevail?
Will this action [QE] tend to depreciate the country's currency against USD because money supply has increased decreasing the value of money?

There is considerable market folklore about this topic. However, from a fundamental basis, all QE represents is a change in the maturity structure of government liabilities. Long-term liabilities (bonds) are swapped for short-term liabilities (money). There is no good explanation as to why that would directly affect exchange rates (ignoring the interest rate effect, discussed next).

From interest rate parity, I believe it would follow that the currency would appreciate against the dollar. If yes, there are two opposing influences on the currency. Which one will prevail?

Interest rate parity is best thought of as an arbitrage condition: interest rate differentials plus the currency basis (which is a traded spread) determines forward rates. This is “covered interest parity”, and it holds (except perhaps in extremely stressed markets).
If interest rates fall due to QE - which is itself debatable - this implies strengthening forwards (assuming the basis does not move). This is necessary to equalise returns.
However, you are asking whether the spot exchange rate will strengthen. From a practical perspective, this is unlikely to even detectable.

The belief that the spot exchange rate will follow forwards is a variant of the Efficient Market Hypothesis, and from what I have seen, this effect is either weak or non-existent. Feel free to search for “uncovered interest rate parity,” it is a large area of research.
The strengthening of forwards is at the tenors associated with bonds. If a 10-year bond yield falls by 20 basis points due to QE, can we really tell whether the currency strengthened by 20 basis points per year over the next 10 years relative to its previous trajectory?