I have a data frame that currently looks like this:
 Month      Park             
   <date>     <chr>            
  2019-04-01 Arbour Lake East   
  2019-07-01 Arbour Lake East             
  2019-07-01 Arbour Lake East                      
  2019-09-01 Arbour Lake East                         
  2019-09-01 Arbour Lake East                       
  2019-10-01 Arbour Lake East                       
  2020-01-01 Arbour Lake East                        
  2020-01-01 Arbour Lake East                       
  2020-02-01 Arbour Lake East                       
  2020-02-01 Arbour Lake East                    
  2020-03-01 Arbour Lake East              
  2020-04-01 Arbour Lake East                 
  2020-05-01 Arbour Lake East            
  2020-11-01 Arbour Lake East        
  2020-12-01 Arbour Lake East                      
  2021-04-01 Arbour Lake East               
  2019-09-01 Arbour Lake West                
  2019-09-01 Arbour Lake West             
  2019-10-01 Arbour Lake West                
  2020-05-01 Arbour Lake West 
I would like to create a new column, Month id, where 1 is the first month found in a specific park, and 2 is the second month in that same park (independently on whether or not these months are actually consecutive. For example, 1 could be September because it's the first month in Nosehill gravel pit Park, and 2 would be November because it's the second month in Nosehill gravel pit Park). Some id (1, 2, 3, ...) would be the same within different parks as they only represent the first month in a park. Months that are exactly the same (month/year) within the same park would also receive the same id.
Here is what I would like that column to look like:
 Month    Month_id  Park                        
  2019-04-01  01 Arbour Lake East   
  2019-07-01  02 Arbour Lake East            
  2019-07-01  02 Arbour Lake East                      
  2019-09-01  03 Arbour Lake East                         
  2019-09-01  03 Arbour Lake East                       
  2019-10-01  04 Arbour Lake East                       
  2020-01-01  05 Arbour Lake East                        
  2020-01-01  05 Arbour Lake East                       
  2020-02-01  06 Arbour Lake East                       
  2020-02-01  06 Arbour Lake East                    
  2020-03-01  07 Arbour Lake East              
  2020-04-01  08 Arbour Lake East                 
  2020-05-01  09 Arbour Lake East            
  2020-11-01  10 Arbour Lake East         
  2020-12-01  11 Arbour Lake East                      
  2021-04-01  12 Arbour Lake East               
  2019-09-01  01 Arbour Lake West                
  2019-09-01  01 Arbour Lake West             
  2019-10-01  02 Arbour Lake West                
  2020-05-01  03 Arbour Lake West 
I really don't know how to do this, so any lead would be really appreciated!
More info:
> dput(Data.frame[1:4])
structure(list(Month = structure(c(18383, 18383, 18414, 18414, 
18444, 18718, 18322, 18687, 18687, 18293, 18293, 18383, 18444, 
18475, 18506, 18536, 18567, 18567, 18628, 18748, 18748, 18779, 
18809, 18078, 18078, 18109, 18109, 18628, 18628, 18444, 18444, 
18475), class = "Date"), Park = c("Aspen Heights", "Aspen Heights", 
"Aspen Heights", "Aspen Heights", "Aspen Heights", "Aspen Heights", 
"Auburn Bay", "Auburn Bay", "Auburn Bay", "Bayview", "Bayview", 
"Bayview", "Bayview", "Bayview", "Bayview", "Bayview", "Bayview", 
"Bayview", "Bayview", "Bayview", "Bayview", "Bayview", "Bayview", 
"Cranston", "Cranston", "Cranston", "Cranston", "Cranston", "Cranston", 
"Currie Barracks", "Currie Barracks", "Currie Barracks"), Aggr_Code = c("1", 
"2", "1", "2", "1", "1", "1", "1", "2", "1", "2", "1", "1", "1", 
"1", "1", "1", "2", "1", "1", "2", "1", "1", "1", "2", "1", "2", 
"1", "2", "1", "2", "1"), AC_events_per_month = c(4, 1, 4, 1, 
2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 4, 2, 6, 2, 3, 1, 1, 1, 1, 8, 4, 
2, 1, 3, 3, 2, 1, 1)), row.names = c(NA, -32L), groups = structure(list(
    Month = structure(c(18078, 18109, 18293, 18322, 18383, 18383, 
    18414, 18444, 18444, 18444, 18475, 18475, 18506, 18536, 18567, 
    18628, 18628, 18687, 18718, 18748, 18779, 18809), class = "Date"), 
    Park = c("Cranston", "Cranston", "Bayview", "Auburn Bay", 
    "Aspen Heights", "Bayview", "Aspen Heights", "Aspen Heights", 
    "Bayview", "Currie Barracks", "Bayview", "Currie Barracks", 
    "Bayview", "Bayview", "Bayview", "Bayview", "Cranston", "Auburn Bay", 
    "Aspen Heights", "Bayview", "Bayview", "Bayview"), .rows = structure(list(
        24:25, 26:27, 10:11, 7L, 1:2, 12L, 3:4, 5L, 13L, 30:31, 
        14L, 32L, 15L, 16L, 17:18, 19L, 28:29, 8:9, 6L, 20:21, 
        22L, 23L), ptype = integer(0), class = c("vctrs_list_of", 
    "vctrs_vctr", "list"))), row.names = c(NA, -22L), class = c("tbl_df", 
"tbl", "data.frame"), .drop = TRUE), class = c("grouped_df", 
"tbl_df", "tbl", "data.frame"))
 
     
     
    