Question about theta in american and european options?

I am assuming you are talking about DEEP ITM (100 delta) options? The reason is the carry rate on the option's premium results in a price less than parity, since a European option can't be exercised into an underlying position. Whereas a deep itm American option won't trade for less than parity, since it can always be exercised into an underlying position.

You also might be noticing some weird things going on with dividends--or perhaps short stock carry rates. If you give me an example I could be more specific.
 
As pointed out above, it's caused by the fact that you cannot exercise a European option prior to expiration.

The minimum value of a European-style put is equal to:

PV(X)-S, where

PV(X) is the present value of the strike
S is the current underlying price

The more time to expiry the lower the PV(X) is, hence the lower the PV(X)-S is. This means that a deep ITM European-style put will trade below parity and as the time to expiration decreases will approach parity.
 
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