I have an 18-mo old now. My company "generously" offers 2 weeks of paid parental leave. I went through similar conversations as you, only I was merely asking to use 2 weeks of PTO after my 2 weeks of paid parental leave for a total of 1 month of time off. My management and HR refused that request, I was told that my option was take the 2 weeks of paid leave and then you can take unpaid leave (the legally mandated FMLA). Due to "project schedule" they could refuse my PTO but legally they were not allowed to deny unpaid FMLA. Truly a despicable amount of leverage. I was also told by my manager at the time how atypical my requested amount of time was and was asked why I needed that much time since I wasn't the one giving birth. I too am astonished at the typical amount of parental leave used/requested by men in the US. It seems like a very deep cultural issue.

I have no loyalty and have just been biding my time. I've been just comfortable enough so far, but things haven't been great in a few aspects. They were not hard up for coverage and my absence, which should have been planned for far in advance, would not have effected the schedule in any manner.