I haven't understood what you said 'Note that increasing the fee for your first transaction invalidates the later (child) transactions'
Assume that you have sent bitcoin from address A to address B and then sent the same coin from address B to address C.
If you bump the fee of the transaction made from address A to address B, the transaction made from address B to address C becomes invalid.
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The total virtual size of your transactions is 427 vbyte and the total fee paid for three transactions is 3953 satoshi.
This means that the effective fee rate is around 9 sat/vbyte.
Your transactions are now around 45 vMB from the the tip and you shouldn't expect them to be confirmed soon.
If you are in hurry, bump the fee of transaction 3.
Currently, for a fast confirmation, the fee of around 20 sat/vbyte is required.
Given the total size of your transactions are 427 vbytes, you should pay 8540 satoshi in total.
You have paid 2945 satoshi for the first two transactions and you need to pay 5595 satoshi for the third transaction. So, you need to bump its fee to 38 sat/vbtye.
Again, if you are not in hurry, just wait.
Edit:
I assumed that transaction 3 is spending the fund received in transaction 2 and transaction 2 is spending the fund received in transaction 1.
The first 2 transactions are marked as 'Unconfirmed Parent' and the third transaction is marked as 'Unconfirmed'. What's the difference, do they link with each other ?
Initially, my wallet have enough fund to make these 3 payments, they should be coming out independently, don't understand why 2 of them are 'Unconfirmed Parent'
I remember I made the third transaction first (the Unconfirmed one), I made the other 2 transactions (Unconfirmed Parent) after that, but not sure which one of the 2 I sent first.