Final answer:
The obstacle keeping Billy from getting the two hound dogs he wants is financial constraints, as evidenced by the quote from the text where he counts his money and realizes he cannot afford them.
Step-by-step explanation:
Obstacles preventing Billy from getting the two hound dogs he wants are financial constraints. This is evident from the text which states, "I counted my money and knew I couldn't afford them."
None of the other options provided, such as lack of space, allergies, or strict parents, are supported by extracts from the text. Therefore, the correct option is A) Financial constraints, which directly quotes Billy's realization of his financial situation impeding the purchase of the hound dogs.
This answer draws from an understanding of the character's circumstances and is backed by the quotation provided.