A convenience function based on behzad.nouri's commend and cs95's earlier answer. Any errors or misunderstandings are mine.
import pandas as pd
import numpy as np
df = pd.DataFrame([["2022-01-01", np.nan, np.nan, 1], ["2022-01-02", 2, np.nan, 2], ["2022-01-03", 3, 3, 3], ["2022-01-04", 4, 4, 4], ["2022-01-05", np.nan, 5, 5]], columns=['date', 'A', 'B', 'C'])
df['date'] = pd.to_datetime(df['date'])
df
#        date    A    B    C
#0 2022-01-01  NaN  NaN  1.0
#1 2022-01-02  2.0  NaN  2.0
#2 2022-01-03  3.0  3.0  3.0
#3 2022-01-04  4.0  4.0  4.0
#4 2022-01-05  NaN  5.0  5.0
We want to start at the earliest date common to A and B and end at the latest date common to A and B (for whatever reason, we do not filter by column C).
# filter data to minimum/maximum common available dates
def get_date_range(df, cols):
    """return a tuple of the earliest and latest valid data for all columns in the list"""
    a,b = df[cols].apply(pd.Series.first_valid_index).max(), df[cols].apply(pd.Series.last_valid_index).min()
    return (df.loc[a, 'date'], df.loc[b, 'date'])
a,b = get_date_range(df, cols=['A', 'B'])
a
#Timestamp('2022-01-03 00:00:00')
b
#Timestamp('2022-01-04 00:00:00')
Now filter the data:
df.loc[(df.date >= a) & (df.date <= b)]
#        date    A    B    C
#2 2022-01-03  3.0  3.0  3
#3 2022-01-04  4.0  4.0  4