Usually I never bring my card out, that's what I ask.
Will not bringing my credit or debit card be a problem to a hotel such as The Park Ave North?
Travelling without a credit in the US is difficult.
It's NOT recommended.
EDIT Hotels and car rental companies mainly want a credit card or deposit to cover unexpected expense that you didn't plan on having in the first place: a car accident, getting keyed in the parking lot (happened to me), breaking something in the hotel room, late night snack attack on the mini-bar, etc.
When you check-in, a card is usually requested, which will be used as a guarantee for incidentals, but also in some cases for the cost of the hotel itself (unless you paid at tile of booking, of course).
Some hotels will simply not accept anything else.
Others will just block any incidentals on your room. That may include locking the mini-bar if there is one, blocking external calls, pay per view, etc, flagging the room for room service, bars, restaurants, etc. so they refuse to charge your room, and so on. Depending on the hotel, this may be very easy (if there are no stocked minibars anyway, no bar or restaurant, etc.) or not.
Others still will ask for a cash deposit instead. Depending on the hotel, it could range from a few dozen USD to hundreds.
Of course, if they also need to charge for the room, there will be no choice but paying in advance or leaving a deposit.
Note that even if you provided your card when you booked, you may still be asked to present the card, as you are switching from the "Card Not Present" regime (used to charge the penalty in case of no show) to the "Card Present" mode. They want to make sure they see the card, and either swipe it or have you use your PIN.
In any case, what exactly happens and what options are available is very dependent on the hotel's policies, so your best bet is simply to call them.
Credit Card not REQUIRED for this hotel
According to website Hotels.com:
Required at check-in
. . .Credit card or cash deposit required
. . .Government-issued photo ID required
. . .Minimum check-in age is 18
It does not specify what the cash deposit is, and is likely tied to the length of your stay (you may need to deposit more than the total bill would be at checkout).