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He sounds interested from what you have told me but as far as the Dog park if you do not live close by and cannot say you were in the neighborhood then I would ask him in passing if he thought it would be a good place to take your dog so that you don't seem pushy if you just show up there. This is the feeling out stage to see if he truly likes you or is just being friendly. Try to see if he shows some of these signs: If he appears around you a lot that could mean that he wants you to see him, and he wants to see you, if he cannot take his eyes off you and want to see you badly then he may be interested so look for that. It's hard to tell if a man likes you because most men aren't into showing outward affection. Men that are trying to show interest in you will tend to look at you when he believes you are not aware of his actions. Yet when your eyes meet, he will look away. He will try to speak with you no matter how trivial the subject may be just to strike up a conversation. In spite of the shyness some men who are interested in you will experience, he will try very hard to appear confident. In fact, he will try so hard that it becomes obvious. This will act as some kind of a mask to hide the way he truly feels. Look for these signs before just concluding that he likes you and wants to get to know you.
I would stop by the dog park and check it out if you only have one visit with him then you don't want to pass up the opportunity and also if he asked you about joining his company maybe you should think about it and give it serious consideration it's not often that you get a chance to meet someone you seem to get along with. I would say check out the dog park it may show him that you are interested in getting to know him and that may make him pursue you more.
Remember to stay positive and take advantage of the opportunity. He may be just as nervous and afraid and confused as you are so remember someone has to make the first move. You don't want to look back years from now and wish you had done something to show him you were interested or live with regrets for not going after what you want it's better to know you tried than to reget not trying at all. Good luck to you! Let me know if you need my help again.