The recommendation for Sunday as the day for baptism is mentioned in canon 856 but without a recommendation that it be within Mass or apart from it.
Although baptism may be celebrated on any day, it is recommended that ordinarily it be celebrated on a Sunday or if possible at the Easter Vigil.
Number 9 of the introduction for the Rite of Baptism of Children says:
To bring out the paschal character of baptism, it is recommended that the sacrament be celebrated during the Easter Vigil or on Sunday, when the Church commemorates the Lord’s resurrection. On Sunday, baptism may be celebrated even during Mass, so that the entire community may be present and the relationship between baptism and eucharist may be clearly seen; but this should not be done too often.
The Rite gives no explanation for not doing this too often, but it is reasonable to surmise that the addition of numerous baptisms on a customary basis during Mass might become somewhat burdensome and hence the caution. Prudential judgment is required.
From my experience in baptizing infants outside of Mass, the rites require a good 35 minutes for even one baptism, but often closer to 50 minutes so that it is conferred reverently and with appropriate catechesis. Other ministers of baptism may have different views on that though.
Since the parents are the ones who present their children for baptism and have an integral role in the rite (indicating the name of the child, asking for baptism, signing the child, being catechized about their responsibilities, rejecting sin and renewing the profession of faith, expressing consent for baptism), it seems that their presence is to be assumed.